Performance/Health checking queries for a user with Read permissions I am a software developer that only had read access to our prod database (of course). 

Occasionally we have what looks like performance issues on our server when my DBAs are not available.

Is there any kind of query I can run (with Data Reader permissions only) that could tell me anything about the current health of my SQL Server?

If there was some kind of edge permission that did not allow for changing data or schema that would help, I could probably get permissions to that.  (For example, I was given permissions to monitor replication in our production environment.)
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.