"What are my options for making sure 2 people won't try to access the database at the same time?"
That's not a viable strategy for a shared relational database application. Not only is it counter-productive to prevent two or more people from working at the same, it is risky. Data can easily be lost in that scenario.
ALL production relational database applications need to be split before deployment to users. ALL of them. Once it goes into use, it has to be split to be reliable and dependable. This is especially important when you are engaged in ongoing development and have to make regular, frequent changes to the interface.
BTW, I can't imagine any case where splitting could decrease functionality. Could you be more specific on that point? What, exactly, did you read which suggested that?
The reason Access can't be reliably run from SharePoint is the differing ways SharePoint operates from the Windows file system. You cannot share an accdb from SharePoint because it can't be opened with sharing enabled in that environment. As you suggest, that means only one person at a time could use it. Highly unproductive. To use it from SharePoint, the accdb must be downloaded to a local Windows Computer, where it can be then opened and updated. All changes are in that copy. That accdb would be uploaded back to the SharePoint site. Potentially, that process would overwrite any work done by someone else who had also downloaded it to work on it during the same time. Oops, last saved wins. And that doesn't include the risk of corruption from conflicting attempts to save the accdb.
There are other, better options to consider.
First, put the back end accdb, containing only tables, in a shared folder on your network. Give each user a copy of the front end accdb, containing forms, reports, VBA and queries.
As you continue to develop it, by the way, this approach makes it much easier to simply replace the accdbs on users' computers with the updated one when you are ready to deploy the changes. The data remains safe in its own accdb, linked to the various front end accdb.
Another option is to export the tables into SharePoint lists. This works, but is not a great solution for larger relational database applications. Again, the front end accdb is distributed to the computers of all users. NO sharing of FE accdbs.
And a third option is to consider a remote-- or cloud -- back end, using SQL Azure, for example. Again, each user has their own copy of the Access FE which is linked to the tables in the SQL Azure or SQL Server database.