Andy,
I'm sorry to hear your experience with other devs. Adding Mootools onto the page is very easy, and we'll gladly help how we can. There's 2 ways to do it, and please note that I know nothing about the other extension, so I can't give you exact things to do, but should be able to guide you well through the process.
If that Vote module on your home page is a MijoVote module (a module that came with the extension), you'll want to edit the following file:
/modules/mod_mijovote/mod_mijovote.php
Please note that I'm making up the names.. maybe it's mod_vote/mod_vote.php or mod_mijo/mod_mijo.php. You should be able to discern the name from the Module manager, hopefully.
Once you edit it, there'll be some copyright text at the top and then you should see something like:
defined('_JEXEC') or die ;
Directly under that line, add another line so it looks like:
defined('_JEXEC') or die ;
JHtml::_('behavior.framework');
If you can't find the right file there, you can do something similar in your template. This will load Mootools on all pages though, which isn't as optimal (but shouldn't be a problem). To do that, edit /templates/<YOUR_TEMPLATE>/index.php and then follow the same steps above looking for the JEXEC statement and adding the line below it.
If that still doesn't work, leave the change there (as long as the page doesn't break some other way) and let us know.. we'll gladly look again and see what else may be wrong.
Finally, again, please check again that you don't have any plugins or template settings that are enabled to forcefully disable Mootools.. as best you can at least
Thanks,
Alex