There are some differences in the runtime.  An Access app built by a
seasoned pro that uses "best practices" throughout would likely execute
under the runtime with no perceptable differences to the user and with
little or no modifications required.  An amateurish app built via the
wizards might not be usable at all under the runtime.  So where between
these two descriptions your app falls is the relevant question.

Assuming a well-built app then yes, you could just install the runtime on a
PC along with your file and then just run it much as you do now.  The
developer's tools will create a setup package to install both the runtime
and your app, but that would only need to be used once per PC.  Future
updates would be a simple matter of replacing the file just as you would
with a licensed copy of Access.

What you need to do is find the help topic (should be available on
Microsoft's web site) that describes what the differences are in the runtime
and what changes you should make (if any).  Also you can just test it on one
of your current PCs by creating a shortcut that opens Access and your app
with the /Runtime command line argument.  This argument simulates the
runtime environment on a PC that has a licensed copy of Access installed.
Then you can see specifically what works and what doesn't.  That shortcut
target would look like...

"full path to MSAccess.exe /Runtime "full path to MDB"

The short list is that the runtime has no access to the built in db window
and no access to any of the built in menus and toolbars so YOU have to
design in all of the tools that allow the user to interface with the app.

In addition, the runtime does not gracefully deal with unhandled errors so
all your code must have robust error handling.  Where a licensed copy would
display a built in error message the runtime will just throw the user out of
the app.  Because they lack error handling it is not a good idea to use any
macros in a runtime app (AutoExec and AutoKey should be okay if they don't
do anything likely to fail).

There is a small handful of features that the runtime doesn't support.
Filter-By-Form being one I can think of off-hand.