Well, Dave, I've got a couple of questions (rhetorical in nature).
First, are the files in fixed format either by columns or by delimiters?
Second, how comfortable are you with macros and modules?
If some or all of the files are fixed in format either of the ways I described, you can manually import each file ONCE and create a named import template. After that, the TransferText method of DoCmd allows you to name a template. Therefore, it is possible to do the import once to build the template and automate it the second and subsequent times.
If the files do not meet either of my fixed-format criteria, your work is a bit harder because a template won't work. You are then facing VBA parsing.
As to how to automate this, two options.
A. Assuming there is a fixed format (by column or by delimiter) for each file such that you can do a DoCmd.TransferText via input template, you can build a macro that does each transfer.
B. If you don't like macros, build one anyway - but then convert it to VBA. You can go back and customize it once it is converted.
Then, whether you leave the result as a macro or as code, create your command button on your intended form but be sure that the control wizard is enabled. One of the options of the button wizard is to run a macro. Another is to run some code. A third option is to lie to the wizard to have it open a form. Doesn't matter as long as that form exists. Then go back into the code editor, remove the OpenForm stuff, and replace it with a subroutine call to the VBA you really wanted to run.
Or cut-paste that code in place of the OpenForm stuff and run the steps in-line if that appeals to you. Six of one, half-dozen of the other.