It's the testicles (primarily) that produce testosterone; a vasectomy leaves the testicles intact but cuts the tube that delivers sperm to the end of the penis, basically.
With no testicles, a trans woman still needs *some* sex hormones, and will have to take estrogen forever. Again, like a menopausal woman, except that the original gonads removed are different (testicles rather than ovaries). She probably will not need an anti-androgen as her body will not typically produce more androgens than the average cis woman, but that still leaves a need for estrogen. Not having any sex hormones runs all the usual risks of menopause, including bone weakening and increasing fragility.