How can we remotely administer our "fleet" of Windows 7 laptops?
Solution 1:
This is the perfect use case for Microsoft Intune. While it is primarily known as an MDM solution, it also has PC management capabilities as well, such as app deployment and patch management. It's also completely cloud-based and is licensed on a per-device per-month basis, so it can scale as you grow.
If you're already managing internal devices with System a Center Configuration Manager, it has some neat integration. If not, it's perfectly functional as a standalone product as well.