Execute a stored procedure as another user permission

As others have suggested you can achieve what you wish using the Execute As clause. For examples of implementation choices take a look at the Books Online documentation for the Execute As clause.

For further reading and to develop more understanding of this topic, what you are looking to achieve comes under the security concept of Context Switching.

Try this:

EXECUTE AS user = 'special_user'

EXECUTE YourProcerdure

REVERT

see these:
Understanding Context Switching <<<has examples of things like you are trying to do
Understanding Execution Context
EXECUTE AS Clause (Transact-SQL)
EXECUTE AS (Transact-SQL)

This is what I did (and succeeded):

let Source = Sql.Database("server", "database", 
    [Query= "EXECUTE AS USER='user' EXECUTE [schema].[spname] 'parm1', 'parm2'"])

in

Source