No, Exact Globe does not have a standard webservice layer. Without any development on the Exact side, there are two methods that you could use to communicate between Mendix and Globe (as far as I know):
Using the webservice component of Exact Synergy to go to Globe through that
Using ASImport
We have done both before.
You could try asking them...? If they have a webservice API we should be able to integrate with it, no problem.