Same issue here

Neither default group nor default event invite box has a search box. 

Did you modify the group/events plugins?
The users appear in the invite box if you have any friends, you can invite only friends. Do you have any friends on the website? 

I have several friends on my site. The invite box appears, but I can't click the search box inside the invite box..... I can't select any users to invite either.....

Eddie, let me note again that neither group not event invite box has search box. It is added by some 3rd party plugin or you customized the software. 

Please disable plugin or revert the changes you did. Check if the issue persists then. 

I've not installed any 3rd party software!

Eddie, if you not installed any 3rd party plugins from the Oxwall store, that mean you did some customizations to the plugins, as default plugins does not have any search in the invite box. Revert the changes you did and then we'll proceed. 

I've not customised anything!!!!!

How come there's a search in the invite box? Default plugins does not have such box. See screenshot

Eddie, invitations are sent via cron, that means that after somebody sends the invitation, it appears in the ow_base_mail table in the database, then when the cron runs ( if it runs at all and properly configured) the table clears out and sends those mails to mail server, as long as the table clears out, software is no longer responsible for the mail. You can check these steps, check the table, check your cron is properly configured

Eddit, turns out that search box is added by Group cover plugin, please disable it and see if the issue with invite persists. 

That seems to have sorted it. Any reason why it's doing that?

Unfortunately I don't know, contact plugin developer to resolve the issue