How can I insert procedure results from server 1 into table in server 2? The goal is to link our website to the database.  The database the website will connect to will be different then our production db.  In fact, a separate instance of SQL.  How can this be structure to execute the procedure in sql1 to populate table A in sql2 so the web query can pull from sql2 table A?
One option is to create a "linked server". This allows you to access another server tables. To do this you use the server name as a prefix.
Before addressing this option verifies with the DBAif  is an option allowed in the organization. Also check permissions issues.

This link from Microsoft has more information. The following example is taken from the same link:

How to create the link?

USE [master]
GO
EXEC master.dbo.sp_addlinkedserver 
    @server = N'SRVR002\ACCTG', 
    @srvproduct=N'SQL Server' ;
GO


How can reference a table from another server?

SELECT local.name AS LocalLogins, linked.name AS LinkedLogins
FROM master.sys.server_principals AS local
LEFT JOIN [SRVR002\ACCTG].master.sys.server_principals AS linked
    ON local.name = linked.name ;
GO