B737-800 Captain reported experiencing unusual vibrations on takeoff roll and they determined a tire had failed. Flight continued safely to destination.

Narrative

During the takeoff roll; shortly before V1; the FO and I both observed unusual vibrations. No abnormal instrument indications or loss of control were experienced; and the aircraft continued to accelerate normally; so the takeoff was continued. The vibrations stopped as soon as the airplane left the ground. Landing gear retraction was normal. After takeoff; ATC (tower or departure) informed us that the aircraft departing after us reported FOD on the runway; apparently from a blown tire. I told them our current intention was to continue to ZZZ. The lack of asymmetry and the normal gear retraction led us to believe the failed tire was most likely on the nose landing gear. This turned out to be incorrect; but also inconsequential.As soon as time permitted; I contacted dispatch via ACARS to inform them of the situation and asked them to contact ZZZ tower for any additional information. I called dispatch on the 'crew phone' feature of the Viasat app to expedite communication. The dispatcher and I both agreed the safest course of action was to continue to ZZZ. Dispatch coordinated for us to be handled as a priority aircraft.The FO and I reviewed the QRH and discussed at great length which landing configuration and techniques would be safest in our situation. I determined that I should perform the landing due to my greater experience with the 737. We decided to land with flaps 30 for best controllability due to the relatively long runway available at ZZZ. Per the QRH; we also decided to not use the autobrakes or auto-spoilers.We also referenced the non-routine landing considerations section of the QRH and briefed the flight attendants and passengers on what had happened and what to expect during and after the landing.We flew a normal ILS approach to runway XXR with ZZZ Fire Command and company maintenance vehicles standing by. Touchdown was smooth; but unusual vibration was felt again immediately after main landing gear touchdown and before the nose touched down. Spoilers were deployed manually; and reverse thrust was used; but the right reverser failed. The aircraft settled slightly to the right during rollout as weight transferred to the wheels. No difficulty was experienced maintaining directional control. I stopped the aircraft with gentle brake application; lowered the spoilers; and asked the FO to extend the flaps to 40 in case of an uncommanded evacuation.ZZZ Fire Command inspected the aircraft and advised us of a single flat tire on the right main landing gear. We shut down the engines. After the brakes were observed to be cooled; company maintenance pinned the landing gear and towed the aircraft to a gate where we deplaned normally using the jet bridge.After all passengers and crew had deplaned and the aircraft was transferred to maintenance personnel; the FO and I deplaned and assessed our fitness to continue duty on our next scheduled leg; which was now late for departure. We determined that due to the high workload and stress encountered on the flight that we were unfit for duty and were removed from the next flight via a fatigue call to crew tracking.Cause: I don't know why the tire failed during takeoff. All tires were observed to be in good condition during preflight inspection and no unusual conditions were noticed on the runway. I'm happy with the decisions we made; and with the support we received from dispatch; ATC; and ZZZ airport. Everything was coordinated well and we had what I think was the best possible outcome.The only thing I feel was missing or overlooked was follow-up from the company to determine if we should continue to operate another flight. It would have been nice to get a call from a Dispatch duty pilot or someone else who could understand the amount of stress involved with this landing and anticipate a need to remove us from further flight duty that day. The fatigue call was absolutely right and necessary; but I don't think it should have come down to us having to make it; at thelast minute; when the plane was presumably boarded and ready to depart as soon as we arrived; and could have felt pressure to continue.