Honest question, should I have taken my newborn for allergy tests before they gave her vaccinations?
Prophylactic natal allergy tests not recommended
Unless you have a family history of various allergies, such testing is usually not done. A familial tendency towards allergies speaks to a difference in the histamine complex on chromosome 6 as an inheritable trait, and unless it's dominant (i.e. everyone who marries into the family without the allergy has children with the allergy), then it's likely not a problem.
That said, a newborn's immune system is largely a legacy from the mother until it's trained up on its own, and the child forms its own T-helper cells which are involved in IgE production, which is where histamine reactions originate. Typically we do not vaccinate newborns immediately for most things (see the CDC site for vaccination schedules for various vaccinations).
Typically, after the newborns own immune system takes over is when you want to vaccinate in any case, since doing so prior to that point will not "train" the newborns immune system to react to the protein you are attempting to train it to react to.
Childhood vaccinations vary. For example, it's common to wait until the child is no longer a newborn to vaccinate for most things. There are also exogenous training factors which should be taken into account, though most are dietary.
If you have a diet containing horse meat (for example), of which the child partakes, then you may find that the child reacts badly to horse serum derived vaccines, such as those commonly used for Tetanus or Diphtheria. You may also find that they would react badly to tratment for exposure to Botulism Antitoxin, which is also derived from horse serum. There was an entire M.A.S.H. episode based on the use of a horse-serum derived Tetanus immunization program in the Korean war that ran into the problem that horse meat was a common component of the Korean diet at the time. This episode was based on historical incidents.
For someone with a large amount of familial allergies, the typical problem is overgeneralization of the IgE response of the immune system. It's unclear whether this is a genetic predisposition, or whether it's a matter of exposure to a large degree of immunological challenges; evidence tends to support a predisposition to certain classes of allergies are genetic (e.g. peanuts, penicillin, etc.), but for them to trigger, you need a second exposure to the allergen after an initial exposure "primes" the response.
We see this same issue present in so-called "Rh babies", where an Rh factor difference between the mother and the baby resulted in the first pregnancy running to term, and spontaneous miscarriages due to the immune interaction with the mother for subsequent pregnancies. It's actually one of the original reasons for state blood tests for marriage licenses, and thus state involvement in approval or disapproval of marriages (the other state involvement reason being the close relation marriage taboo, which until allele testing was possible, didn't show up on the blood tests the way a mother/father Rh factor mismatch would).
I believe the best common practice for this, if you are concerned, is to sit down with your child's doctor *well before* it's time to make a "yes/no" decision -- i.e. don't do it on "shot day" -- and discuss the issues and your concerns and family medical history with her or him; most frequently, the rational decision, when you aren't forcing it in the heat of the moment, is usually to do the vaccination.