I know the Membership provider stores the user name and an expiration time in an encrypted cookie and then uses that to verify the user is still logged in for a session.
Would it be possible to store the users password in this encrypted cookie as well. If so how would you access it server side?
I need the users username and password available server side because I need to call web services that use those same credentials. Is there some better way to do this?
You should store it in session state, which never leaves the server.
You should also try to change those web services to use authentication tickets instead of passwords (eg, OAuth), because it's never a good idea to store passwords in plain text.