Within the last few years the laws regarding obtaining passports for minor children have changed to help reduce the number of international custody disputes/abductions. The law now requires that for any minor child to obtain a passport, both parents must appear together in person with the child to submit an application.
You state you have sole physical custody, but the question is whether or not you have sole legal custody and even with sole legal custody, it is possible that the passport agency will not allow you to obtain the passport without your daughter's other parent being present or without an explicit court order which states the other parent's legal consent is not required or that the other parent is not a legal parent of your daughter.
If you can, just ask your daughter's other parent if he/she will come with you to obtain the passport. Or if you already have an order which states any of the above, the passport agency should recognize your right to obtain the passport. If you do not have a court order which states any of the above, if the agency refuses your application, or if you can't obtain the cooperation of your daughter's other parent you should consult with a skilled lawyer to assist you in obtaining the passport.
Keep in mind however, that even if you have a valid passport for your daughter, and sole physical custody, you will still need your daughter's other parent's permission to travel internationally for each and every trip. Airlines and customs officials are requiring notarized consent forms when one parent travels alone with minor children, even if the parents are married or otherwise in an "in-tact" relationship. If you are unable to obtain consent from your daughter's other parent for the specific trip, you may need to obtain a court order to make sure that you will not be stopped by the airlines or customs officials during your trip.
It may not be necessary to get a lawyer, especially if you are able to get the cooperation of your child's other parent; but, it may be a good idea to have a lawyer review your current custodial arrangement to advise you more fully and to ensure your trip happens smoothly.
Sign up to receive a 3-part series of useful information and advice about child custody law.