The MC Server can't add my server?

I made an agreement with someone to sponsor my MC server.

They have decided to make the Sponsors Area so that when you click on a button it will connect to my server via BungeeCord.
They have added my IP address (in number) and the port.

For some reason they cannot connect to my server. My server is not BungeeCord, so is the problem because my MC server isn't BungeeCord?

My server is not an VPS. It is from a host. How do you fix that?

First of all, your server doesn't need to run Bungeecord. Bungeecord is a proxy between different servers. You connect to the bungeecord server (e.g. 10.0.0.1) and this server bridges the connection with another server (e.g. 10.0.0.3 or any ip)

Is their config.yml set up correctly? Is it in any way possible, to show the config here?

Have you checked out the bungeecord configuration guide? Perhaps this could help you and your sponsor. If you have any other questions, I'll be glad to help.

edit:

Oh, and my server is not an VPS. It is from a host. How do you fix that?

Your server does not need to be hosted on a virtual private server, the bungeecord instance connects players to your IP.