Does this application require authenticated network access to other systems? If not, this might work. It's a variation on AlexIT's solution, if more tedious.
75 machines is a non-trivial amount, but you might be able to get away with making a new local user on each of the machines and adding that user the the Administrators group on the local machine. Make sure to give it a password only you and trusted staff will know. Regular strong password policy should apply, since this is an admin account, and a savvy user could still take advantage of it should the password be compromised.
Go into the local security policy (Administrative Tools) on each machine and open 'Local Policies' -> 'User Rights Assignment'. Find the item for 'Deny logon locally' and add the new admin user you just created. This will limit users from trying to logon locally to the machine as this user.
If this is an application that needs authentication to the network to use resources, you might have to make a regular user on the server which is added to the local admin account on all of the workstations. Restrict local logon as above. I would suggest against making an additional network level admin. While it would save time to simply add another user to the Domain Admins group, it's a little too risky to do that just to get an app to work when it puts your entire network at risk too.
After that, set up custom shortcuts to 'Run As', like AlexIT has listed above. On 75 machines, I don't envy you the task of visiting each one, but you might be able to mitigate some of that with scripting.
One final note that I thought of. If this app needs individual user authentication to direct some level of access within the application itself, you might be unable to go with the 'Run As' approach, since the application will either think that it is connected to by one network user (which was added to each local admin group) or else a bunch of local machine users that happen to be local admins too, which won't mean anything server-side. If this is the case, you might be doing as AlexIT suggested, and getting in touch with the software vendor too find which registry keys and files need the appropriate permissions. I've done this before too, and sometimes it can be as simple as giving the "Everyone" group access to a temp folder. Your mileage may vary.
I may be doing this 'Run As' workaround enterprise wide (800+ machines) in my organization over the summer. Some apps just don't want to work without admin privileges (or a ridiculous amount of work on my part), but giving admin rights to local users is simply becoming too risky. You aren't alone in this. We're all doing it.
