If you opened up the firewall from the service provider to your Exchange Server for port 25 or used a VPN, yes.
Then create a receive connector for their application:
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We have a server running Exchange 2010 and one of our service providers want to use it to send their application e-mails, but the application isn't on our network at the moment.
is it possible to allow a web server to use the local mail server to send it's emails?
If you opened up the firewall from the service provider to your Exchange Server for port 25 or used a VPN, yes.
Then create a receive connector for their application: