There is no online version of full AutoCAD/LT: but all subscribers have access to a limited ability cloud service called AutoCADWEB
https://web.autocad.com/ , and all active subscribers can also install AutoCAD/LT on a second PC if the owner of the license (if not you) will allow you that privilege.
Access to your office's DWG files are determined by how and where you store your files and how you make them available outside the office, that has nothing to do with the Autodesk software. Consult with your IT if you don't have something already set up.
HTH