This is a sigh question, of course you can. No one stops you wearing whatever you wish when you have a spray tan.
There are no laws, lol, but every piece of jewellery you wear will be spray tanned too.
Your gold will be spray tanned, your silver the same and your diamonds and other precious gems will be spray tanned.
Plus wouldn't it just be better for tan lines if you just took it off? I'm pretty sure the people who do spray tans advise you to take jewellery off.
When we get to vintage costume jewellery, some of these are delicate pieces and you are just going to coat them in chemicals, that will cover the stones and the metals, and probably decay the settings as well as the metals and stones.
Of course you can choose to wear your jewellery whilst being spray tanned, if you want outlines left when you take jewellery off. I guess that could give a pretty interesting look for a while. If that's
what you are after.