I think there is no doubt that the answer is "yes". The question is whether they are particularly successful at it, or not; and whether they have to supplement tournament winnings with other activities like teaching.
In addition, a common use of the term "professional" in sports is if someone gets paid, and the Grand Prix consists of four events with prizes in each ranging from 3,000 Euros to 20,000 Euros, and the overall prize standings at the end from 2,500 Euros to 15,000 Euros for the standings winner. So they do get paid, and thus, are professionals, at the very least, technically.
That said, I think that most may have their participation in tournaments and income supplemented beyond teaching, most probably by national federations, but organizers may also offer incentives to attract top women players. The Grand Prix pays the women's travel expenses, and lodging (Section 5.2), for example. Some may even pay modest appearance fees, but these would surely be few, and far between.
WGM Natalia Pogonina's manager, Peter Zhdanov, wrote here that "Saratov State Law Academy helps WGM Natalia Pogonina by paying for some of her chess trips". Again, it always seemed like she is, or was, one of the "professionals", but how successful was she? Based on what Peter Svidler told me about men a long time ago, I would guess that it is a rough life financially without some kind of outside sponsorship.
Things have changed for men, but around 20 years ago, Peter Svidler told me on ICC that with the exception of world championships (same for the women today as the recent women's world championship had a prize fund of 500,000 Euros), at that time, even the top 10 men only earned around $50,000 per year. Of course, at that level for men, you really cannot do it part-time, and supplement with lessons since it requires that much dedication. Judit Polgar, clearly, did nothing but chess, and in 2015, Hou Yifan only earned an estimated $265,000.
While it is relative, these women are at the same level as the men, meaning at the very top when competing against their own gender, and I cannot see how they would be doing it part-time any less than a man could in the top 10. Most of the women in the Lausanne Grand Prix are grandmasters, and to be a grandmaster, for most, is a full-time occupation at the "top".
Of course, this is all discussing being a professional primarily from tournament winnings. The Internet today makes it easier for anyone, even weak players, to start a Twitch channel, and develop a following, and earn a living if they are good at attracting an audience.
P.S. I found it very difficult to find hard sources with quotes about how top women chess players sustain their careers, but I did put what I could find, and the rest is just common-sense analysis from reading over the many years.