Is the vasectomy effective birth control right away?

No. There is a delay until vasectomy can be relied upon. Your partner will need to have a follow up semen analysis to confirm the absence of sperm.
Up to 99.85%... Vasectomy will not provide an instant contraception so post-vasectomy precaution against pregnancy has to continue until post-vasectomy semen exam shows no sperm. Besides, recanalization may occure. Note: its effectiveness for birth control is up to 99.85 % although post-vasectomy re-canalization rate up to 1-5%.
Not for some weeks. Vasectomy prevents sperm from going into semen, and should include: the surgery, using other birth control or condoms to collect the next 20 ejaculations (for 6 weeks), and then giving a semen sample at the urologist's office to be sure no more sperm is in the semen. After all three of these things are done, a vasectomy is effective (extremely low failure rate of approx.1 in several thousand men).