Seeking free web GIS browser?
If you are not looking to build your own, and rather use a hosted web service, there are a few that are out there to consider:
QGIS Cloud - A potentially easy option if you want to maintain your data in QGIS.
MapBox - Read the following question for options of converting QGIS projects to TileMill: Are there tools to directly import QGIS projects into Tilemill?
CartoDB - A great option if you want to move your data to PostGIS.
A webgis solution seems ideal for your problem.
The client will be able to log to your server, view the data (based on the permissions you gave them), do basic operations, and its many-to-one solution.
Here are some open source web-gis platforms you want to consider:
- Mapbender
- Geomajas
- GeoMoose
- MapServer
- GeoServer
If you want to avoid writing any code then MangoMap is your best bet. It's also free which is a bonus!
Just upload your shapes, update the styles, press publish and you're done!
Disclosure: I am the Founder of MangoMap