From my reading, 5e uses the "Great Wheel" interpretation of cosmology by default, but I'm having a hard time understanding what this means. Under this model, are the planes physically connected, and could a Githyanki pirate bring an airship to the Material Plane (without the use of teleportation magic or astral projection) to rob people?
No and no:
No, 5e does not use the Great Wheel cosmology — not by default. It's one of the many options available for a DM, but 5e presents only ingredients for a homebrewed cosmology, and very few pieces that are assumed as defaults (usually in things like magic item descriptions, such as the bag of holding).
In the Great Wheel cosmology, the Astral is entirely separate from the Prime Material. It is only accessible via magic, either more powerful magic that directly moves one to/from it, or via smaller magic that can get you into the Ethereal, through which you can travel to get near its border with the Astral and then use other magic or portals to hop from it to the Astral.
This means that, no, a Githyanki ship can't raid outside the Astral without very powerful magic that can essentially Gate the entire thing. Even then, an Astral ship only works in the Astral Sea anyway, and would need entirely different magic to operate in a Material atmosphere/space/crystal sphere/whatever arrangement that particular world in the Prime has.
Basically, the planes are physically separate and can't be traversed by mundane means. There are some exceptions (such as the borders between the Outlands and the Outer Planes, but even that's complicated, since how the “border” works between those infinite planes is quite strange in practice), but as a rule: no, you can't just walk, fly, sail, crawl, or trebuchet between the planes.
On the plus side, this means that you don't have to figure out why Githyanki don't constantly raid the Forgotten Realms or somesuch.
But on a completely different plus side, the lack of a default cosmology means you can trivially say that the Planes are contiguously connected in a useful way to allow Githyanki raiders to bother the Realms, if that is the campaign you wish to construct!
The planes are separate, independent and infinite. There are connections between the planes but, in general, they are not spacial connections that you can walk, sail or otherwise move between by physical means.
The DMG describes planar travel starting on p. 44:
- Planar portals - these can connect any plane to any other plane as decided by the DM. Sail your airship through a portal (assuming it is big enough) and there you are. Portals can (and should be) weird and wonderful: random destination, random times, only usable by certain races, on certain phases of the moon, permanent and stable, shrinking, growing etc.
- Plane Shift - from here to any plane you have the right fork for. This is, of course, the origin of the often used expression "fork you" meaning, in essence, "I hope that you are sent to another plane which is not to your liking." It only moves up to 9 creatures so you have to leave your airship behind.
- Gate - this creates a portal from here to any plane you like, it can be from 5 to 20 feet in diameter - you could fit a small airship through this.
- Astral Projection - from any plane to the Astral plane and from there through portals to any other plane. Again, only 9 creatures so no airship for you. In some cosmologies not all planes touch the Astral plane so it doesn't work from those planes.
- Etherealness - from the plane you are on to the border Ethereal but no further.
The particular cosmology is relevant only to the extent that it determines what planes there are and which planes touch.