Who "owns" a project idea after a grant application is rejected or declined?
If you can prove you are the originator of the idea with timestamped emails, notebook with dates, journals, etc. then that may be sufficient to prove priority for the idea when registering intellectual property (IP).

If the collaborators furthered the idea with you, then they own part of the idea, and may make a claim to any IP that is generated.

If you worked on this idea / project while you were being paid by University X, then University X may also own partial rights to the idea/IP.

Keeping the previous in mind, anyone that contributed to the idea could use the idea in another proposal.

If you are the originator of the idea, you can submit the idea to another group. If intellectual property is produced from the project, others that helped you develop the idea (including University X) could try to claim partial rights. You are not required to inform prior collaborators that you are using the idea, although it is a courtesy to ask them to collaborate on the new project if it is accepted.

Normally, grant application ideas are somewhat vague since you rarely know what you are going to find during the project. No one "owns" this idea until it becomes intellectual property through patents, trademarks, etc. That is when you really need to consider who contributed.

Note: Intellectual Property rights differ in every country, and Universities have very different policies. Check your local legislation and University contract.