Creating a flier team on limited resources?

Based off your options and my experience (with all the fliers out now, I have like 4-5 flier teams lol) I'd say your best bet is L!Ryoma, Cordelia, B!Ninian, and either Cherche or Camilla, depending on the type of green unit you're looking for. I'd try and snag the new Nino if possible to solidify the green spot, but it's understandable if you don't want to toss orbs at the banner. Shigure is another possible slot you can consider, but I'd want to keep Bridal Ninian on there for dancing capacity. I'm partial to them myself, especially on fliers, but if you feel you can keep them safely positioned without, it could free up a slot for another damage unit.
I'll start with the obvious that you're probably aware of, but I don't see any of the hone flier units. In that case, you'll want to hoard Palla's for goad fliers. Since the buffs stack, it'll give a noticeable performance bump with proper positioning. I'll break it down by unit after this, but flier teams tend to abuse positioning advantage, heavy buffs, and often use firesweep weapons, so you'll want repositions all around, and any firesweeps you can. Always helps to have at least one.
Ryoma's base kit is strong. His B skill adds guaranteed damage, which is always good against armor teams, especially when they stack ward and you can't damage them. If you can manage a firesweep sword, and don't mind losing his DC effect from Raijinto, you could pair it with Life and Death (as high as you have). His already insane speed will stack even higher, and you can get safe damage in with firesweep and bushido. Pair it with moonbow in case you run into any physical walls. His sacred seal should either be heavy blade or quickened pulse, but that also relies on which one you give to cordelia.
Ninian is mostly support, but a damn good one at that. She doesn't quite take the spot from New Year's Azura for me, but offers a great kit. Dancers are mostly personal preference. I prefer wings of mercy, but it's usually not needed with fliers. Her chill atk could save you if one of your units has to take a hit (wish it was chill spd or def though). For higher investments, I prefer blade tome since it works with armor buffs, and she's typically my only magic attacker on her fly team. If not, just refine her default weapon for speed. Any buff seal works, like hones, since she is a support primarily, and hones stack with goads.
Cordelia, you'll really want a firesweep on. Really changes how fliers play imo. I know her galeforce build is popular, and that's the one you'll want heavy blade for (seal or A skill if you can manage, but probably seal). Galeforce is great, but my problem is the lack of extra damage from a special. I like my firesweeps to at least have the potential, and Moonbow helps me chip away at armored units at least. But both have their merit, so I'll leave it to you. She should get life and death if you have it, but death blow or fury works if you don't have spares.
Green is a tough spot for me, mainly because I have ex luck and am probably a whale (combine the luck and small whale status, and I prefer to view myself as a narwhal). My main flier team runs Azura in green, my secondary uses Myrrh, I now have the new Nino, and Bridal Sanaki is a beast. So I usually run a broken mage, or sometimes just pass on the green slot and use Kinshi Hinoka or Bunny Kagero. Now, with you silently judging me, we will explore the options you do have available. Cherche is pretty straightforward in that she runs the default brave axe build. Brave Axe+, Luna/Glimmer, Death Blow, and hit and run. Simple, and effective, just don't let her get touched by mages. If that's your style, go for it. On this one, though, I prefer Camilla, especially with her new axe. Makes her build slightly less expensive than the slaying axe+, distant counter build that some of us built when we were bored (surely not I...). Get Camilla's axe, take the effect refine, and you have a great flier support green. I run mine with DC still, but a fury/desperation combo should do nicely with her.
Hopefully, this helps, and if you have any more questions, feel free to share and I'll answer them to the best of my ability. I think you'll enjoy fliers, though, with the proper investment. They've always been the most fun imo since they're highly mobile glass cannons half the time.

Look, I don't want to burst your bubbles but you'll never have a fully functional flying team without hone fliers. It's a tough skill to get but until then you can do something.
Ryoma is a good unit. If you wanna use his distant counter (Raijinto) you may want to buff his poor defenses. I'd heavily consider Camilla for the job. She is a bad unit but have a GREAT weapon refinement. She gives +3atk/spd that stacks with whatever flying buffs she carries. So it's a good thing to fodder Palla to her to get Goad fliers. She'll give flying allies +7/+7.
Next you may take shigure that is a powerful unit that can be built offensively, defensively or mixed, depending on IV's and skills. His base kit is decent enough and he benefits from Camilla's attack buffs. Shigure have a Defensive skill that you may use or switch to Caeda's Fortify Fliers.
For the last you decide: can go for cherche to brute strength or Cordelia to Brave attacks. A red unit would be the best but Palla is really not worth it. As soon as you get a unit with hone fliers you rearrange your team for it.

There you go, my budget flying emblem.

@Leao Goad Fliers is a thing. If you don't have hone, get all goads on them. This allows you to even take on panic ploy units without having to be positioned out of its way first. Pretty good if you want to straight up blitz the enemy, which flier teams should be doing anyway.

As for teams, if you're ever thinking about flier emblem 5* merge projects, Caeda, Cordelia, and Camilla should be your core team and they can be super budget. Just merge as they come. Caeda having dual effective weapon allows more leeway for your offensive options. Cordelia's Brave Lance set is already super good but you can also switch to Firesweep Lance depending on your team's needs. Camilla and her new weapon refine allows her to boost herself and fellow fliers even more on top of goad fliers, which makes her a very strong flying support unit.

I have a tendency to try to squeeze every last bit of utility out of anything. I consider myself a good candidate to attempt to make a team out of your members.

Several observations:
- as someone else noted, no hone fliers. This isn't end of the world, as that responder suggested. Goad stacking is still extremely powerful.
- no Caeda = no fortify fliers. Your team is going to have a hard time tanking when you get into awkward or tight spots (one example might be facing a cavalry team on that arena map that has a bunch of destructible rubble in the middle and then it turns into an open field, or the indoor map that has a bunch of destructible rubble in a checkerboard pattern so that the cavalry team opens it up at their leisure but you can't just reposition into them and strike first because the rubble is in your way)

B.Ninian: dance. The rest, I defer to the expertise of others.
L.Ryoma - if you're trying to go minimal SI, for now play him with Iote's shield seal, reposition, and if you have enough Palla, goad fliers. You can inherit the firesweep set onto him later if you like that option. A-skill probably wants to be replaced; also consider giving him desperation if you're not planning to switch to the firesweep set. After you tank a hit or two you go desperation attacking.
Camilla: get the effect refine on her axe for extra buffs, which will help you get around the desire for hone fliers. She will need to tank the blues. I recommend to dump her res because she has enough. When Camilla first received her new weapon I did a super-budget build with her by grabbing reposition and spur atk 1, 2 from a Barst; goad fliers, night sky, and moonbow from Palla; draw back from Donnel because I'm not in shortage of the guy. (QR is an option if you're swimming in Subaki, but you might not be forever and there's a good chance it's not your long-term option so think twice.) And I kept darting blow; with goads stacked on her she has enough speed to block doubles and darting blow gives her the extra speed she wants when attacking. In the absence of Caeda I'd save Clair if I were you (rather than foddering hit and run) because you might want to play Rhomphaia at some point.
Cordelia: my preference is firesweep galeforce. Heavy blade seal and L&D is good; if you dance her she should have no trouble getting 3 turns every player phase. You will want to attack with her a lot if you go with my other options because I've given you last-resort tanking options at the expense of some aggression.

With this precise team, the person who wants to supply fortify fliers to the team is Cordelia. She abandons all defense to attack, so others can pick up the defensive buff and use it to reposition her out when her turns are exhausted.