Couldn't find anything online about this, but I'd like to give a SQL Server role access to a linked server using a specific login to that server. I know you can assign a specific local login to a linked server login with sp_addlinkedsrvlogin but I'm not looking to do this for every login (nor am I looking for a script to do it for every login). I suppose there's some security reasons to prefer the use of an account, but I'd like to know if it's possible.
Not exactly possible, no.
To manage group-wise access, you're supposed to use the Be made using the login's current security context option (doc here). Then, if you're authenticating with Windows authentication, configure the login and user and security access on the remote server with a Windows group that your users are a member of. (If you were using SQL authentication, you'd need to have a login with an identical password on both servers to use this context.) Then you'd have to create the role on the remote server, assign permissions to that role, and assign the group to that role.
This is kind of a pain, but it does mean that no matter how your Windows users connect to the second server, they'll get the same access.