Some pregnancy tests (as far as I know most/all) use animal antibodies, called monoclonal antibodies. These antibodies are grown/farmed from animals, often mice or rabbits.
Because these pregnancy tests use animal products, I don't think they count as vegan (though that's somewhat subjective). Are there any clearly vegan tests?
Tracking of your basal body temperature as described in detail in the lower half of this article - https://www.verywell.com/body-basal-temperature-chart-to-detect-early-pregnancy-1960284 Is an old-fashioned method of identifying a pregnancy which would avoid the use of animal antibodies.
Further scientific source: https://www.mayoclinic.org/tests-procedures/basal-body-temperature/basics/definition/prc-20019978
For more information regarding the proven correlation between basal body temperature and ovulation/gestation: https://www.webmd.com/baby/tc/basal-body-temperature-bbt-charting-topic-overview
Also see: https://baby-pedia.com/charting-body-temperature-pregnancy-change/
If you're willing to be patient, and don't need an answer ASAP, there's always the ultrasound. An ultrasound can show the gestational sac as early as three weeks after conception (the fifth week of the pregnancy). Using a transvaginal ultrasound, the gestational sac can be detected up to a week earlier. The fetus itself can usually be detected a week later, and the fetal heartbeat, while not audible on a fetal doppler, can be seen as a flutter on the ultrasound around 6.5 weeks into the pregnancy. While there is delay compared to when a dip-stick style test could detect the human Chorionic Gonadotrophin (hCG) hormone, which might be able to detect the hCG during the second week of the pregnancy (almost a week before the expected period), it is very reliable.
If a dip-stick method detects a pregnancy, the next step is usually an ultrasound anyway. The process of scheduling an ultrasound, especially with the way some insurances work, it's likely that the time-frames will be right for detecting the gestational sac, at least. As a side benefit, there's nothing quite so reassuring as seeing the flutter of the fetal heartbeat on the screen while they're doing the ultrasound, and most technicians will also print out a copy of "baby's first picture" for you. A dip-stick test offers no such benefits.
Late to this but the answer is a solid no if you're looking for commercially available Beta-HCG tests. According to this guidance by the FDA:
Presently, all of the home pregnancy tests available use monoclonal or polyclonal antibodies in an enzyme-linked immunoassay format.
And getting into the merit of production, all antibodies need animal cells to be produced at one point or another - you could argue that monoclonal antibodies are "the lesser of two evils" (so to say) since the production is mostly done via cell culture, but the "blueprint" for the antibody came from an animal that was injected
