Embraer first officer reported while taxing for departure experiencing both main gear tires being flat resulting in shutting down the aircraft waiting for maintenance action.

Narrative

Aircraft X was scheduled to depart ZZZ at XD00 with 5 passengers on Day 1 at XD00. The night before; the captain scheduled a rideshare for a XB45 pick up from our hotel. This gave us enough travel time to the FBO for our XC00 show.On Day 1 I woke up at XA45 and was down in the hotel lobby at XB30 eating scrambled eggs out of a paper cup. The captain joined me shortly thereafter and told me that his scheduled ride was cancelled; the rideshare app was unable to find a driver.He asked me to try looking on a rideshare app for a driver. I started looking; and I also called a few local taxi companies. He did the same; and neither of us were able to find a ride to the FBO. The captain called the FBO to ask for a ride. Part of this phone call included him communicating with the FBO about fueling the aircraft before we arrived. I am not sure exactly what was said; as I was still trying to get ahold of cabs. The FBO agreed to pick us up. The hotel is about 15 minutes away from the FBO. The captain called scheduling (to the best of my memory) to let them know that we would be late. On our way to the FBO; we saw a vehicle on the road ahead of us that the FBO employee informed us was our passengers. We arrived at the FBO around XC30; about the same time that our passengers did. I noticed when we arrived to the FBO that the fuel truck was hooked up to the aircraft. The fueler remarked to me that he was unable to reach the amount of fuel that the captain had requested. I realized at this point that there had been a misunderstanding between the PIC and the FBO about how to fuel the aircraft. (Normally the door to our fuel panel is locked. Earlier in the tour and before I was paired with him; the PIC had lost his fuel panel key. I have one; and he tasked me with the locking and unlocking of the panel. I must have failed to lock the panel after our last fuel stop in ZZZ1 on Day 0. Additionally; it was not locked at shutdown when we parked in ZZZ on the night of Day 0.)I completed all of my normal and preflight duties per our AOM/FOM.The aircraft had more fuel on board than our release allowed; so the PIC called dispatch for a new release and he updated the passenger and bag weights from standard to actual. We received a new release. The PIC went inside to get the passengers. He loaded them and I loaded their bags. He gave them a safety briefing; and part of his briefing included the fact that we had been overfueled and would need to park somewhere and burn fuel before we could depart. We were cleared to taxi to XXR. During our taxi to XXR; the tower offered us XYL; which we accepted. The PIC back-taxied down the runway as directed by tower. I remember noting the time as XD00 during this taxi. I did not notice anything unusual about this taxi. We parked in the pull out next to the departure end of XYL. We told tower of our intentions to burn off fuel. The PIC set the parking brake and advanced the thrust levers to expedite the fuel burn. A few minutes later the aircraft began a slow rolling turn to the right. The PIC reduced the power to idle; stopped the aircraft with the toe brakes; and re-engaged the parking brake. He then advanced the thrust levers again. A few moments later we achieved the level of fuel required to depart; and i notified tower that we could take off.Tower cleared us for take off and the PIC released the parking brake. As soon as the aircraft began to taxi it was clear that we had an issue. The PIC stopped the aircraft and I told tower that we had an issue. Tower cancelled our takeoff clearance. The PIC and I agreed that we probably had a flat tire. The PIC elected to taxi clear of the threshold area for XZL. He very slowly and carefully taxied a distance of about 50(ish) yards. He started making phone calls to owner services and maintenance and asked me to talk to the passengers. I told the passengers that we had an issue and would be unable to depart. I told them that a van from the FBO would come getthem. I told them that I would keep them updated on the van and on their recovery flight. I don't have a perfectly clear memory of the next series of communications that occurred. Both of us were talking to the FBO and to tower. The PIC was talking to Ops and maintenance. I was keeping passengers updated. Tower was very kind and helpful; we told them we were unable to move the aircraft. The FBO van showed up to take our passengers to the FBO. My best guess on timing for this would be about XD40. At this point we shut down the engines and unloaded the passengers. I assisted them and unloaded their bags. I noticed that both main gear tires were flat. The passengers were taken to the FBO.The PIC and I had a discussion; during which he told me he had been riding the brakes on the taxi to XZL. I had not realized this was happening. Tower advised us that the aircraft could stay where it was. We secured the aircraft. At XE20 the FBO van returned to the aircraft again to pick up the PIC and I. Suggestions: Familiarity with proper de-fueling procedures.Inclusion of SIC on all actions intended to be performed on the aircraft. Familiarity with recommended procedures in the event of thermal plug release on a wheel. Strict adherence to SOPs regarding locking of fuel panels and only fueling an aircraft with a crew member present.