The answer is No. You do not need to give the TOEFL test at the time you apply to the Fulbright. Only successful (Principal and Alternate) Fulbright candidates will be required to take the TOEFL. Candidates are notified of the success of their application in October after their interviews, which are conducted in September. So you will be asked to take the TOEFL somewhere in November/December.
In case you have already given the TOEFL within the last two years, then you don’t need to take the TOEFL at all. There is no minimum TOEFL score requirement for Fulbright – it actually depends where the Fulbright and IIE end up applying on your behalf. If they requirement of your programs is 90, then that would be your target. To be safe, try to keep your TOEFL above the 100 mark on the IBT exam.