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Well, I've answered my own question! Here's how you do it:
First off, the search form needs to include those elements that you need to send to the submit page to make it work. In this case, all I need is the forum elements present on the standard XOOPS search block, along with the additional checkbox element that tells the search page to limit searches to my module. This additional element has it's display set to 'none' so as not to confuse the user.
First off, add a statement like the following to the php page associated with whatever page you need your module specific search on:
$xoopsTpl->assign('thisModuleID', $xoopsModule->getVar( 'mid' ));
Second, add the following code to the appropriate template page that the above php uses to generate the html for the user:
<form style="margin-top: 0px;" action="<{$xoops_url}>/search.php" method="get">
<input type="text" name="query" size="34" />
<input type="hidden" name="action" value="results" /><br />
<span style="display: none"><input type='checkbox' name='mids[]' value='<{$thisModuleID}>' checked='checked' />module name could go here, but we don't care since it's hidden anywayspan>
<input type="submit" value="Submit" />
form>
That's it! I hope this helps someone out. Feel free to use and/or post this code however you like!