Or did they just not want to be second?

Have they even orbited the moon with a manned spacecraft? Why are they stalling?

I belive they orbited an unmanned probe around it, but thats it.

The Russians are the masters of one way space travel, so, no.

kursk...

They did, but the N1 booster that would have gotten the upgraded Vostock/Soyouz type capsule and lander into low Earth obit for departure to the Moon kept blowing up shortly after launch, or right on the pad.

They definitely did/do have the tech, but they were saddled with procedural and bureaucratic issues that made it difficult to respond effectively to engineering flaws in the designs. They gravitated towards designs that had dozens of smaller rocket motors, which increased the number of failure points, more plumbing, more controls etc. more complexity IMO, whereas NASA and the Apollo program contractors threw all their effort into the Saturn V's five MASSIVE F1 engines and making them absolutely FLAWLESS, which are still engineering marvels to this day.

The situation was somewhat reversed when the Russians had MIR and simple and reliable Proton launchers and Soyouz capsules, and we were limping along with the fragile and overly complicated Space Shuttle.

Rocketdyne really did an amazing job with the development of the F-1, they even came up with an F-1A, which was more powerful, yet 33% lighter, and was possibly slated for post Apollo space-station building projects, and Mars missions with the Nova booster, but the contraction of NASA under Nixon, and the unrealistic promises of the Space Shuttle program killed it.

The Russians actually had the first soft landing on the moon (Luna-9, 31 Jan 1966) and had a bunch of probes make to the moon––and back.  Granted, there's a huge difference between sending a manned vs. unmanned mission to the moon, but I definitely think that they've proven their ability to put A) man in space for long periods of time, and B) put objects on the moon and bring them back.  There's no money or political advantage in sending people to the moon, so they don't bother.

The technology: absolutely
The means: no

The greatest design in the world will not get off the ground if you don't have enough money to pay people to do it right.

Bet they could send another dog or monkey on a one way trip.

Really who returned the USA astronauts when the last shuttle blew up?

They really only lacked a capable booster. In a very odd role reversal the US beat the snot out of the Russians in the heavy lifter department with the Saturn V. They didn't even consider that an engine like the F1 could be made, so they got stuck with a 32 engine first stage. Their lunar lander was a very small thing with no room for future capabilities but it flew in earth orbit and it would have worked. Had Korelev survived long enough to oversee the N-1 booster development the Russians may well have beat us to the moon. They didn't have the desire or means to do anything more then get there and walk around, but they could have done it first.

Today? No, they are even less capable then the US of reaching the moon. As much experience as they have in low earth orbit, thats all they can do right now. The US has the benefit of a more developed aerospace industry, lots of deep space probe experience and neat toys and techniques from the shuttle program. For the US it's really a matter of spending the money to build the stuff we've designed, for Russia it's a matter of spending money they don't have to develop the technology and techniques they don't have.