I am new to SSL certificates and trying to install one from https://letsencrypt.org with certbot.
I see that I need to prove I own the domain by putting a file accessible from the web that to hosting running that domain.
When I get the certificate, will it be somehow locked to the hosting I use or can I just put the files somewhere on my computer then upload them once I deploy on this or that hosting environment?
No, absolutely not. Once you get the certificate you can move private key and fullchain/server certificate anywhere. Move it to another server which is going to host that domain/SAN(subject alternate name)