If I set a script to run as admin, does that mean any user that has the script run will not need permissions for things done in the script such as transaction search?

Run the script as current role

If you run the script as "administrator" it will do as suggested and the user role will not play a part. If you want the script to be restricted to only perform tasks that the user would normally be able to perform then set as current role as suggested above. Note you may want to look at the deployment options too and ascertain the scripts scope with regards to departments / users etc.

Can it be done with client script?

No. You can limit the script running based on who is the current user but not assign a role to the client script like you can some server side scripts. See the deployment on the client.

Ok, I think I know what to do to fix my issue. Thanks for the help!