If you buy a business and want to just own it but not work in it, then no permit is requried.
If you want to actively work, opinions vary on whether you "need" the permit or not.
But getting such a permit makes life a lot easier than trotting off to immigration every 30 days, so it seems like the sensible thing to do regardless of whether it's a strict legal necessity.
Owning a business would put you in the "Self-Employment Permit" category - this is generally quite easy to process. It's just a matter of paper-pushing.
Be aware that there are times when elections, scandals, etc. cause a back-up or slow-down in paper-processing.
But in general, also remember that Belize WANTS investors, so if you are a legitimate business person this can be a great place to "do it".