Are you sure? As far as I can understand - and from how it's explained on that thread I linked to - it's simply using Google Custom Search.
OK - because I've been known to get things wrong, I thought I'd better check this out myself.
I copied and pasted the code I quoted earlier into a page on my website, and found that it did indeed work. Anything I typed in returned a page of results from the UK vintage forum. That proved that Google isn't simply looking at the referring site before deciding which site is raising the query...
Next, that funny business in the "cx" field:
I changed a single character in that string, and that stopped the search from working. Good - as suspected, the long string in the "cx" field is indeed telling Google which site is being searched.
So I decided to set up a Custom Search, using the URL provided by gmb42 earlier. That was dead-easy - literally just a case of typing in my site URL and clicking on "go". In return, I have been given a load of stuff that I can paste into my website to generate the Google search:
<script>
(function() {
var cx = '005351797449373264470:qjxoqys2r60';
var gcse = document.createElement('script');
gcse.type = 'text/javascript';
gcse.async = true;
gcse.src = (document.location.protocol == 'https:' ? 'https:' : 'http:')
'//www.google.com/cse/cse.js?cx=' cx;
var s = document.getElementsByTagName('script')[0];
s.parentNode.insertBefore(gcse, s);
})();
</script>
<gcse:search></gcse:search>
But, this all looks a bit complex, so I haven't tried that yet. Instead, I took the HTML that I posted earlier, and simply replaced the "cx" value with the one given to me in the above code. And it works!
All of that took about 5 minutes. Adding something like this to the forum ought to be dead simple - it's just a case of understanding how SMF organise their files/templates/whatever, so that you know which file needs to receive the HMTL to put the search box onto every page...
Good luck!
Mark