Referencing a referenced entity's fields on a form
You can use DFV - Dependent Field Values. It's a great module and does exactly what you're asking
The module's page says:

This modules enables site admins and developers to make the allowed values of a field depend on another proviously selected field's value.
You can use views with arguments to let views return the allowed values for a dependent field.

I'm using it myself, it's quite simple to use: you create a view of type Dependent Field Value with a contextual filter, which in your case is the NID of the node in the first field, the team (remember to add the relevant relationship in the view's Relationships section).
Usually you'll want to add two fields to this view, e.g. the home game's NID and the home game's title. After adding the two fields, click on the settings of the  dfv list format. Choose the home game's NID as the Key Field an the home game's title as the Title Field. The view is done!
You didn't explain how is your contact form made, but I suppose that you use webform. So: you just have to go to the second field's settings, check the &quot;Enable dfv settings&quot;, choose the parent field (team) and the view you just created.