In Office 365/ SharePoint Online I came across and issue where I would share a site with an external user, but the invitation would not appear in the "Access requests and invitations" of the site setting. This did not prevent the invitation from being sent or the users accessing the site, but it made it difficult to track the invitations.
As it turns out, this is a know problem, but the cause was unknown. If you have this problem, then Microsoft has a script they can run to fix the issue. You will need to contact them to run the script in their hosted environment.
After going through a number of scenarios, I determined the root of this issue. If you create a site template from a site you created, you can create new sites from that template and the invitations work properly. If you create a new template from a site that was created from a template, the Access request and Invitations will not display your invitations. You will need to call Microsoft for a fix.
To avoid the problem, always make your changes on your original clean site and make new templates from your model template.