Can you update a database with transaction logs only, without having to do a full restore?
I would like to run a process that loads in the latest log file every hour and then run a full restore once a week (when we receive the latest bak file). Is there anyway of doing this?

Assuming you run the same version of SQL Server, Yes.  What you're looking for is called RESTORE WITH STANDBY:

RESTORE WITH STANDBY Leaves the database in a standby state, in which
the database is available for limited read-only access. This option is
equivalent to specifying WITH STANDBY in a RESTORE statement.
Choosing this option requires that you specify a standby file in the
Standby file text box. The standby file allows the recovery effects to
be undone.

Every time you apply another log backup users will need to be disconnected briefly.  This is how Log Shipping works, and so long as they send you all the transaction log backups, you don't need to restore the weekly full.  Just apply the next log backup in the chain.