Yes, you can always request original equipment manufacturer parts after you've had an accident. Some insurance companies might make you pay extra for OEM parts.
Most insurance contracts obligate the insurance company to restore your vehicle to the same condition it was in before the loss. Sometimes this requires the use of original equipment manufacturer (OEM) parts and sometimes after-market parts can be used. After-market parts are parts made by a manufacturer other than the original manufacturer.
Some states have laws concerning OEM parts. Most states permit the use of after-market parts as long as they are warranted by the manufacturer to be of like kind and quality as OEM parts, but you don't have to accept them. The final choice is yours but if the insurer wants to use non-OEM parts and you decide to use more expensive OEM parts, you may have to pay the difference in cost.
State regulations typically require the insurer to clearly indicate in writing on the appraisal which parts are after-market parts.