Georgia law doesn't require having a fixed address to vote. You need to maintain GA residence. The full text of the relevant statute is here[0]. Note section 15b.
The form requires an address for contact and precinct information. People without a permanent address can use a common place of residence (where they can receive mail), such as a shelter, or if they don't have one of those, they can register in the park they sleep in [1].
Ultimately, it is up to the district registrar to determine if the residency is legitimate, but this is decided not based on the existence of a conventional fixed address, but based on the demonstrated intent to reside in the state (and more specifically the municipality).
[0]: https://law.justia.com/codes/georgia/2010/title-21/chapter-2...
[1]: https://faq.georgiavoter.guide/en/article/how-to-register-to...