Can a person carry a virus in their body for years before it comes out?

Yes, it is possible, for virus AIDS, you can be positiv without to have any manifestation, at these moment you can alwready give the virus to someone else.
With herpes, once you have it it is always there even if you do not always have the manifestation, you are contagious only when they are manifestation. It can be that it doesnt comes out for years and then it can come back sudenly...
A last example with Chikungunya these mosquito disease , you carry the virus some years after you're safe.
I guess there are a lot of exemples.
you can carry it befor, like aids, and after also for other disease

after each disease is different, i guess
but it is definitively possible.