A low white blood cell count can in fact increase the risk of infection. If this is the case, you should see your primary care doctor
as this requires urgent attention.
In general, the white blood cells are the part of the blood the fights infection. Different types of blood cells fight different types of infections. If your white blood cell count is low--this can cause recurrent infections. This is known as being immunosuppressed. This is a major problem that requires attention. There are acquired causes of immunosuppression (including steroid use, leukemia, HIV) and there are inheritable causes as well. If your white count is very low then you should have this worked up as to figure out why this is occurring.
In general, there is a wide range of normal white count. People even with the lower end of normal white count are not at increased risk for infection. Talk about what your white count is with your doctor.
To answer your question, there is no evidence that taking extra vitamins will increase the white count. There are some vitamin deficiencies like folate and Vitamin B12 that when low can effect blood cell development, however taking extra is not proven to be of benefit.
I would talk to your doctor regarding both your frequent infections (which could represent a serious disorder) as well as your folate and Vitamin B12 level. These can be checked by blood tests. Good luck.