Performance/Health checking queries for a user with Read permissions
Have your DBAs create 


Adam Machanic's - sp_whoisactive 
Brent's - sp_AskBrent


and grant execute permission to you.

That should be a good starting point to find out why your sql server is slow.

Note: Brent has already mentioned on How to Grant Permissions to Non-DBAs. You can follow that method to ask your DBA to grant permissions to you.