What happens when a transaction says 'Pending' in Airbase but the vendor says it was declined?

When a charge is declined, your card would’ve been charged and hence it will show up in Airbase as Pending. The charge will be dropped by the Vendor and when it does, it will be moved from Pending to Declined in Airbase. 

For Bill Payments, if the money was debited from your account, it'll be credited back to your account in 6-7 business days.