I trained pest control professionals:
The "how" of entry on an RV is more about the tiny openings that allow them to get inside. They easily climb jacks, hoses, wires, tires, and so on. But finding where they actually make entry may take some careful searching.
The "good news" is that explorer ants usually lay down a scent trail for other rank-and-file ants to follow. That means that, if you can find the early parts of the trail...perhaps on a stabilizer jack...you may be able to follow the ants as they use the trail and see where they are getting in. If you find the point of entry, you can seal it up with caulk. That's a permanent solution.
Virtually all pests come for three things: 1) food; 2) moisture; 3) a place to live. In an arid climate, moisture is a powerful attraction. I think we can rule out shelter as a motive - except in rare circumstances.
Assuming there is a way into the plumbing via your gray-tank dump, it's plausible that they entered through your gray tank and then followed their way to the sink. The problem with that hypothesis is the drain trap. Rodents can swim through a trap, but ants usually cannot. Furthermore, if they are in your gray tank, they have LOTS of food and moisture. Anything is possible, but it's far more likely that they are in the sink to take advantage of moisture remaining near the drain, and they got there via a route inside your RV...perhaps their trail is through the vanity cabinet.
The good news about many ants - especially the tiny "pavement" ants - is that you can disrupt their scent trail simply by washing it away with soap and water. You MIGHT be able to reduce the attraction of moisture by drying out the sink after use, but nothing will stop them from going down into the trap from the sink to get the moisture they need.
A chemical pesticide will be effective at killing and repelling them, but bear in mind that pets and little kids are VERY vulnerable to these chemicals. The old saying is "the dose is the poison." The exposure of a 150 pound man is very different than the exposure of a 14 pound lap dog or 40 pound child. And pets might find the smell of a pesticide worthy of sniffing or licking. Whereas you would instinctively not "huff" the scent of the chemical, a dog might sniff in overdrive attempting to identify the smell.
Also,"long lasting" residual pesticides ("Protects for 6 months!!") are meant for long-term protection. If you camp in multiple locations over a summer or move frequently, repeated applications (dosing and redosing) to your running and landing gear and connection systems leads to quite an accumulation of the toxic chemicals on your RV, and it leaves "contamination" for those who occupy the site after you leave. You also get exposed repeatedly as you apply it, handle shore-power wires, hoses, stabilizer jacks, and so on. If you handle contaminated surfaces, wash your hands ASAP after handling these items.
It's a stretch to mention this, but most states require a license to apply pesticides on another person's property...or a commercial property - e.g. a campground. Applying pesticides on an RV campsite may be a violation of the law. Again...a stretch. If, however, you were to discover an infestation of fire ants on your site, it would be unwise and probably illegal to tackle it on your own. What you do in your own home is up to you, but it's better to report something like this to the campground owner and ask to change sites. Leave them to hire a professional to handle the situation. It's also likely campground staff would be in violation of the law if they were to attempt to do a pesticide application themselves. These laws vary from state to state, but the EPA govern's much of this.
A better option? Consider alternative repellents. Many rely on disrupting or overwhelming the same acute sense of smell the ants rely on to communicate and to locate food/water sources.
So, an infestation is one thing. That means they've taken up residence in your RV...as carpenter ants might in a moist wall structure. But your situation is more of an opportunist expedition for food and water, and you can start by sealing up entry points and then using repellents to deter them from trying to get in.