There really are too many variables on this one:
Answer me this
1, How many tables and records in them?
2. How much contention for the rows?
3. How much experience in programming do you have?
4. How much experience in networking do you have?
5. Any web experience?
The simplest solution would be to copy a read-only access database to all 3 locations and have them use microsoft access to look at it. This would allow the folks to look up information but not add or modify any of it... While not strictly useful it may satisfy what they need to do.
A middle solution might be to put the access database on a shared drive all 3 locations can hit and enable it for read-write. I would do a backup of it every day and you would have to lecture everybody about contention (what happens when 2 people are modifying the same record)
One of the hardest would be to create a MYSQL database on a hosted linux server and access it using a custom PHP solution. It is the most challenging because if you don't know MYSQL, SQL, PHP that is a lot to learn.
Also never forget they will want to generate reports out of this system. If they tell you that they do not, THEY ARE LYING!!!
Best of luck