Airtable is almost exactly that: a collaborative, simple MS access. It's free for up to 1,200 records, then $10/user/month for more https://airtable.com/pricing
Thanks! I've considered it before but not sure anymore why I did not choose it. Though it probably relates to the fact that we are a small volunteer project with insecure funding (i.e. currently none) that tries to keep a records database of contact information up-to-date. This goes hand-in-hand with complex permission management/input controls (i.e. those listed should be able to submit a request for modification or for adding their record) and the possibility for integrating the result in a website. I don't think such integration tools are included in the free version
Why don't you just ask them if you could use it for free? It's not much cost to them and they could do something good.
Another vote for Airtable. It's truly a game changer.