Most vaccines are not troublesome for people with egg allergies, even though many vaccines are cultured in eggs. The two vaccines people with egg allergies are most concerned about are the MMR and the flu vaccine. Here's how to proceed before getting these vaccines.
Of the vaccines on the regular CDC schedule for children and adolescents, the only one that is potentially problematic for people with egg allergies is the influenza (flu) vaccine. That even applies to the MMR vaccine. While the MMR is cultured in eggs, studies have shown it to be safe for people with even severe egg allergies. The CDC, American Academy of Pediatrics, and the British Department of Health "Green Book" all recommend that the MMR be given to egg-allergic children without special precautions.
The flu vaccine is also cultured in eggs and presents a slightly higher risk of reactions than the MMR vaccine. Because of this, the ACAAI (American College of Allergy, Asthma & Immunology) recommends as a best practice that people with egg allergies be given a patch test to the flu vaccine itself. In the event of a positive test, the patient and her doctors will decide whether the benefits of the vaccine outweigh the potential risks of a reaction (such as when the egg-allergic patient is in a very high risk group for complications from the flu). If doctors do decide to give a flu vaccine to someone whose patch test came out positive, they will do so in a setting where emergency treatment, like epinephrine, can be given immediately in case of a reaction.
The other vaccines on the regular CDC vaccination schedule -- DTaP, HPV (Prevnar), Meningococcal, Pneumococcal, Hepatitis A and B, Polio, and Chicken Pox -- have not been associated with reactions in people with egg allergies.
Do ask your doctor about the possibility of reactions to eggs in any non-standard vaccines you may receive in unusual circumstances or before traveling. The yellow fever vaccine is cultured is eggs and may cause reactions, and is required for travel to some countries of sub-Saharan Africa and tropical South America. Your egg allergy will allow you to obtain a medical waiver from a travel clinic, but you should discuss alternate ways to prevent yellow fever with your doctor before your trip.
Sources:
American College of Allergy, Asthma & Immunology. "Flu Vaccine & Egg Allergy." Patient Education Fact Sheet. 21 Nov. 2007.
Fox, Adam & Gideon Lack. "Egg Allergy and MMR Vaccination." British Journal of General Practice. Oct. 2003 53(495): 801-802. 19 Nov. 2007.
CDC. "Vaccines and Preventable Diseases: Who Should NOT Get Vaccinated With These Vaccines?" Patient Education Fact Sheet. 11 Dec. 2007.

