Should citations be included in funding applications
Yes, include references. They serve a twofold purpose in your application:


They inform the reviewer about the state of the art in your field, and allow him to judge whether your project would advance it in a meaningful and cost-effective way.
It shows the reviewer that you know what you are doing. Nothing worse than a grant application for a project where the PI did not read the literature thoroughly and does not know that his exact question has already been thoroughly investigated.


The funding body you apply to may well have very clear-cut guidelines for your application. (Because applicants have a tendency to write up everything they know about the field they are excited about, and no reviewer wants to read a 200 page proposal.) These guidelines will very likely have a strict page limit and explicitly note whether references should be counted against that page limit or not, along with what citation style to use.

If there are no guidelines for references, use whatever is common in your field, e.g., in common journals or conferences. Use conventions that make reading your proposal easy for the reviewer.