Hi,
It sounds like you essentially want to make your database accessible either to customers, or whilst colleagues are out of the office.
I've been building database solutions for the UK's largest retail bank for over 10 years now, and so have a huge amount of experience in this field.
I'd suggest rather than trying to connect your existing database to the web, you actually move the whole database there.
A MySQL database and PHP based front end could sit on your current linux server without any additional licensing costs. I would use the Bootstrap framework to build the GUI, making it responsive so that it works extremely well on both mobile devices and full size computers.
Should you choose too, you can also connect directly to the database, allowing you to interrogate the data using Access as you currently do.
Please feel free to message me to discuss this further, I think this could be a great benefit to your business.
All the best,
Ben