You start by saying you are commissioned to build an Access database and then you proceed to tell us that you cannot use Access. You need to clarify what exactly your client needs.
Access is a Rapid Application Development (RAD) environment. It is used to build client/server applications and with A2013, web applications that run under SharePoint. Jet/ACE are the desktop database engines used natively by Access. Access can also link to any data source that supports ODBC connections so Access is also frequently used as a FE to Oracle, SQL Server, DB2, etc. "Access" is a separate product from Jet and ACE. Except for the fact that "Access" uses Jet or ACE (depending on the Access version) to store its internal objects, "Access" is not welded to Jet/ACE. If you were to use C++ to create a client/server application that used a Jet/ACE database, you would not even need Access to be installed because you are not using Access. You are only using the database engine - Jet/ACE. You use "Access" when you build forms/reports/etc. using the MS Access GUI.
Jet/ACE are DESKTOP database engines. They are FILE servers rather than true database engines and although they can be used as the BE for a web application, I wouldn't recommend it. They will support only a small number of concurrent users and will not perform as well in the "cloud" as SQL Server will.
If you are not using Access to build the FE, you are not using Access!!!!! You are using Jet (.mdb) or ACE (.accdb) which are separate products.
You need to clarify exactly what the remote connection needs are. Is this a "want" because someone thinks the web is cool and client/server is passe? If outsiders will be accessing your web page, Access and Jet/ACE are certainly the wrong tools. If the app is inward facing where everyone will be on the same LAN or can use Citrix to get into the LAN, then in reality, they will probably be happer with a client/server app.
If your client doesn't want to incur the upfront cost of creating his own Sharepoint site, you can host one in the cloud. You'll need a small business Office 365 account to create the A2013 web app. That runs around $16 per month. Please verify because you will need to know if you have space limits and remote connection limits. Then individual users should be able to get by with the basic $6 per month plan. You might have some number of free connections but I'm not sure they can work with your Access database. The whole setup is pretty fuzzy and I haven't seen a site where it is clearly defined.
If you go the Cloud route, your "Access" database will be using SQL Server Azure as its BE rather than Jet/ACE or Sharepoint Lists. As far as I know, web pages are not resized for small devices. If you want them to be usable on small devices, you will need to create forms specifically for that purpose.
There are a couple of other downers regarding A2013 web apps -
1. NO VBA. You are stuck with macros.
2. Macros can't do everything so make absolutely certain that they have the functionality you need before you begin.
3. There are no reports in the Access sense. They are only long web pages with no page breaks.
4. There is no charting or pivot tables, etc.
5. There is no integration with Outlook, Excel, or Word so make sure you won't ever need to import/export/ or email.
6. Form design is pretty rigid and you have no artistic options.