If the Flyboard Air meets the definition of an ultralight from 14 CFR 103.1 then only part 103 regulations would apply. 103.7 says that ultralights are exempt from airworthiness, registration and pilot certification requirements, and 91.1(e) says that general flight regulations don't apply to ultralights.
Based on the specifications on Zapata's website, there are two models: the Flyboard Air EXP and the Flyboard Air UL. The EXP doesn't meet the part 103 requirements (its fuel load and top speed are too high) but Zapata says the UL was designed to comply with part 103 and the quoted specifications do seem to be OK.
Assuming that you're fine with the UL model there are still two points in part 103 that might be an issue for you. First, the stall speed requirement in (e)(4):
(4) Has a power-off stall speed which does not exceed 24 knots
I don't know how the FAA would define a "stall" for a Flyboard and their detailed guidance on part 103 (dated, but seems to be still current) doesn't clarify it. It does say that a "recognized technical standards committee" can confirm that the requirement has been met, but I don't know who that means. You might want to ask Zapata how the UL complies with this requirement (and how they documented it).
Second, you can't use an ultralight in some commercial activities. The FAA document lists activities that are allowed (e.g. airshow participation) and others that aren't (e.g. aerial advertising). If you plan to use it only for private recreation then that doesn't matter, of course.
Finally, note that the FAA's document says that if you "encounter" an FAA field inspector for any reason it's up to you to prove that the ultralight meets the definition in 103.1. If you can't, they'll consider it as an aircraft for compliance/enforcement purposes. Before making your investment, I'd make sure that you're confident you can prove it really is a part 103 ultralight. If you aren't sure that you have the right information to do that, you could contact your local FSDO and ask them to review whatever you have and tell you if they would be satisfied.
(This answer is only about federal aviation regulations. If the Flyboard causes a lot of noise as mins mentioned, or disturbs people or the environment in some other way then there could be non-aviation, state or local laws that would apply.)