Install and set some free and spatialyspatial enabled database which is supported by all GIS software like
- PostgreSQL with PostGIS or
- MS SQL express
Upload all your TAB files with MapInfo EasyLoader into the database. After that you can access your tables from QGIS and MapInfo and database will handle multi-user editing.
Installing databases and setup will require some specific knowledge that you and your team maybe dont have but you have a zillion tutorials on the Internet.