Microsoft Sql Server - How to replicate a table from Linked Server?
It doesn't sound like any of the built-in SQL features are going to meet your needs here:


SQL replication makes a lot of changes to the source db, and probably won't work over a linked server
Mirroring and/or always-on availability groups likewise require lots of underlying configuration to the source server
Log shipping, if you had access to transaction log backups, would bring over the entire database state, not just the tables you want


You're going to have to either use a 3rd-party software (like Redgate SQL Compare), or roll your own. 

Personally I'd build an SSIS package to pull down new or updated data from the required tables (hopefully you have some easy "last_update" fields to go off of) and run it via a SQL job every few minutes. There is no SQL wizard for this, you'll have to know what you're doing or hire someone who does.