Most online communities have an e-mail a friend feature. It’s probably not used much.
It’s too much effort to find your friends’ e-mail addresses.
You need to simplify the process.
Don’t ask for e-mail addresses, ask for names and any details that will help find the person. e.g. place of work or location.
Now you can use Google, LinkedIn and Facebook to track the person down and send a personal invite. If you’re really clever, you can write a code that does this for you.
If you can’t change the coding of your website, then start a thread every 6 months on the forum. Invite members to list any friends with the most basic of details to help you find them.