I'm creating a campaign for my group and I need to know if dragons (metallic and chromatic) can enter and exit the Feywild, without having to use fey crossing portals?
Any dragon of CR 21 or greater could potentially know Plane Shift innately
The "standard" spell for crossing into arbitrary planes is Plane Shift, a 7th level spell. Any dragon that could cast this spell would be able to use it to access the Feywild. The Monster Manual's section on dragons presents a variant titled "Dragons as innate spellcasters", which says, among other things (emphasis added):
Dragons are innately magical creatures that can master a few spells as they age, using this variant.
A young or older dragon can innately cast a number of spells equal to its Charisma modifier. Each spell can be cast once per day, requiring no material components, and the spell’s level can be no higher than one-third the dragon’s challenge rating (rounded down).
Since Plane Shift is a 7th level spell, any dragon whose CR is 21 or higher could potentially know how to cast it. Furthermore, dragons do not require material components to cast their innate spells, so such a dragon would not need to acquire the costly and hard-to-find material components needed to cast the spell. So, a sufficiently powerful dragon interested in visiting the Feywild would be able to learn to do so using their own innate spells.
So, which dragons have a CR of 21 or greater? Well, the highest CR among all adult dragons is 17, so that leaves only the ancient dragons (greater than 800 years of age). Among the ancient dragons, most have a CR of 21 or greater. The only exceptions are white and brass ancient dragons, which have a CR of 20. So any ancient dragon that isn't white or brass has the potential to know Plane Shift and thereby access the Feywild on their own.
One thing to keep in mind is that any dragon with the ability to cast Plane Shift has access to any plane, not just the Feywild. Given this, it's unlikely that a dragon with interest in the Feywild would not at least dabble in visiting other planes as well.
Of course, Plane Shift is only one of many ways to access the Feywild. There may be other ways for lower-CR dragons to gain easy access, and even if there aren't, as DM you are empowered to create such ways in order to serve your world and story.
There is nothing in official 5E material that suggests they can. In fact, (most) dragons are not of the type "fey" — they are their own creature type:
Dragons are large reptilian creatures of ancient origin and tremendous power. True dragons, including the good metallic dragons and the evil chromatic dragons, are highly intelligent and have innate magic. Also in this category are creatures distantly related to true dragons, but less powerful, less intelligent, and less magical, such as wyverns and pseudodragons.
There is a shadow dragon — kind of a template for transforming a dragon touched by the Shadowfell — but to date no sort of equivalent for the Feywild. There's fairie dragons, and while those are definitely kind of fey-like, they're not actually of the fey type either, and really are a different sort of thing. In any case, none of these point to a general planar affinity, let alone one to the feywild.
There is an optional rule which a DM can use, in which dragons can innately cast spells no higher than 1/3rd their CR. The spell Plane Shift is 7th level, so if you use this rule, any CR 21 or higher dragon (some but not all Ancient dragons) could shift to not just the Feywild but to any plane. However, with this, they couldn't shift at will — each spell can only be used once per day, so there-and-back-again is at minimum two days.
That said, you note that you're creating a campaign. Awesome. You're not constrained to the variant rules listed in the Monster Manual. Those are there to help you build your world, not to hold you back. It sounds like you have an idea that centers around dragons and some connection to the Feywild. Check out the Dungeon Master's Guide Chapter 1, A World of Your Own. That chapter begins:
Your world is the setting for your campaign, the place where adventures happen. Even if you use an existing setting, such as the Forgotten Realms, it becomes yours as you set your adventures there, create characters to inhabit it, and make changes to it over the course of your campaign. This chapter is all about building your world and then creating a campaign to take place in it.
As the dungeon master, if you want dragons to have this ability, you should absolutely give it to them.
This isn't just an exhortation to "think outside of the box". It's absolutely what D&D is about! From the intro to the Monster Manual:
Naturally, you can do with these monsters what you will. Nothing we say here is intended to curtail your creativity. If the minotaurs in your world are shipbuilders and pirates, who are we to argue with you? It’s your world, after all.
Giving dragons in your world a link to the Feywild and corresponding abilities including travel to that plane is exactly in line with this. Add these powers, and follow the guidance in the DMG to help you think through the implications of that change. This is a great place to start world-building!
If you prefer to follow something a little less-free form, consider "Creating a Spell" from DMG chapter 9. There's unfortunately not much guidance for non-damaging spells, but do note that Plane Shift is particularly powerful: it will let you get to any plane (at least if you have the correct planar tuning fork), and serves as a sort of multi-realm teleport, taking you right to your destination. One can imagine a lower-level spell allowing access to the Feywild with greater constraints (including, say, no ability to target unwilling creatures). You could make this available to anyone, including dragons, either using the existing variant rule or by tweaking it a bit to bend the CR/spell level restriction.
But personally, I like the idea of just giving dragons this ability. That makes the world more uniquely yours.