Budgets are posted to a ledger so I do not think there is a good way to delete them, rather just upload and release a budget with zero values for any account/subaccount pair you wish to reset. 

Thanks

@CJohnson15 It depends on what ACU/PC sync engine you use.

If you use the Acumatica-built one,  then you will need to break the link between ACU/PC using the Project Sync screen but be careful because it will break all links for the selected project entities including Budget, COs, Subcontracts, …

If you use Procore-Built sync, then you will need to reach out to PC support as it will need to be done inside Procore if possible. 

I am still using the Acumatica built one. Can you tell me how to get to the project sync screen? Do you mean the ProCore integration or is this something else.

Thank you so much in advance for your help!

It is part of PC integration module but I believe it is Sync Cost Budget. You can not do it without removing links of subsequent entities. If you want to break link at once for everything, you will need to go to Sync Projects and break the link for your desired project and as mentioned in my first reply it will break everything so be sure of it. 
if you have not received training on Sync engine, then have your partner train you because the sync engine related forms (I guess there ears around 9 screens in total) is a bit clunky in my opinion.
As a general guideline, you break the link from exact same screen you created the link.