The burning question no one ever asks. The answer is yes. If anyone believed they couldn't, they probably fall for similar misconceptions such as legendaries are genderless (proven false), legendaries are unique (see Pokemon Snap), and real world animals are replaced by Pokemon (untrue). Why? Because people take what they can and can't do in the trainer games too literally. In that case, bikes are also the most expensive thing on the planet, and when lightning (which on average is about 30,000 amps of power and can reach up to 50,000 degress F, give or take) strikes rocks or any natural non conductor, they will always remain unscathed.
I don't see how people can get so hung up over debating this. Official sources themselves tell you they can, and elementary biology can
clue you in on the rest. It doesn't even require thought. What do people have against the Nidos that they don't want them to be able to breed? Any mother will tell you that having seeing that child, that she has just given birth to, for the first time brings about an immeasurable amount of joy. Why take that away from mothers?
First of all, let's take the major games as they are. Nidorina's Pokedex info in Gold states:
When feeding its young, it first chews and tenderizes the food, then spits it out for the offspring.
Nidoqueen's Pokedes info in Silver:
It uses its scaly rugged body to seal the entrance of its nest and protect its young from predators.
Already the game contradicts itself, because in the very same generation that started the egg business, the information clearly states them caring for children. This isn't even the only instance. Nidorina's Pokemon Stadium 2 information repeats this, and jjust about every Pokedex entry for Nidoqueen since has stated the same thing (probably due to a lack of inspiration to write anything interesting, but that's neither here nor there).
I know someone's thinking "well, it's possible that the children are grown up and the Nidorina/Nidoqueen had them when she was a Nidoran". Fair enough. Indeed it's probable that Nidoran gained enough experience just trying to survive to evolve into Nidorina after having birthed children. However, a freak moonstone incident soon afterward? That's honestly reaching, and I'm even taking into consideration that moonstone dust might be in the air (based on the Pokemon episode Whichever Way the Wind Blows). I'm also taking into consideration just how often we get rocks from the moon.
It wouldn't be a stretch, however to believe that when and if Nidorina evolves, that Nidoqueen would want to have children, or more children. Animals can and usually do have more than one litter of kits within their lifetimes, mostly wild animals, in order for their species to thrive. Same is true for Nidorina, given that there is no age requirement per evolutionary stage.
If legendary birds can technically have children, I don't see how a lesser species of rabbit hybrids would be any different. Apparently, the anime already knows this.