After the fall of the Soviet Union there were serious concerns both about the control of its massive nuclear stockpile and its massive stockpile of fissile material (bomb fuel). To put it into perspective, the USSR had tens of thousands of nuclear warheads at the end of the Cold War (either deployed, in its stockpile, or awaiting dismantlement) and it had many hundreds of metric tons of highly-enriched uranium and plutonium (it takes about 60 kg of HEU to make a crude bomb, and about 6 kg of Pu to make a weapon but the weapon is harder to make).

Soviet command and control procedures were pretty loose — it was not easy for the USSR or later Russian Federation to certify that they had not lost weapons or fissile material. Many facilities lacked even basic inventory procedures that were common elsewhere in the industrialized world (e.g., bar-codes and databases), and security was especially poor in some remote areas. The difficult insecurities, both economic and political, that followed in early post-Soviet Russia put an especially fine point on this since the people guarding these facilities were making essentially poverty wages (and thus might be tempted to "augment" their income).

The US approaches to this included the Nunn-Lugar Cooperative Threat Reduction Act, in which the US paid the Russians to upgrade their nuclear security, and the Megatons to Megawatts program, in which the US bought surplus Russian plutonium and blended it down for use as fuel in US civilian nuclear reactors.

Anyway. To your question: is it plausible that nuclear weapons or fissile material were stolen in the early 1990s? Yes, it's totally plausible. The weapons would be presumably a harder commodity to steal and use. While not all Soviet weapons had sophisticated locks on them (Permissive Action Links), there are indications that many of them did. Whereas 60 kg (130 lbs) of HEU is in theory something you could smuggle out in small pieces (and HEU is very hard to detect with radiation detection equipment — it is easy to shield, it is not very radioactive).

But did these things happen or hit the black market? There have been no public indications of serious sales of fissile material, much less weapons, on any kind of black market. That doesn't mean it hasn't happened and wasn't discovered (or was discovered and just shut down quietly), it doesn't mean that interdiction efforts had a perfect rate of success, etc. While any details about this sort of thing are no doubt quite classified, it is clear that the US and many other agencies (including presumably the FSB, Interpol, etc.) are very concerned about the idea of terrorists acquiring nuclear weapons and have no doubt used a variety of means to attempt to detect such things. There have been several cases of people trying to sell radioactive materials, or people claiming to be selling HEU on the black market (but it turned out to be a scam), for whatever that is worth.

I don't know about biological munitions (I am not sure the USSR had any stockpiles at the end of the Cold War — they had developed a capacity to produce stockpiles but I don't think they had stocks on hand other than small samples) or chemical munitions (no clue). But in the nuclear realm, yes, it's plausible. Did it happen? We don't know. We don't have evidence that it did happen, but absence of evidence is not evidence of absence.

If you are interested in Soviet WMD endeavors in the late Cold War, David Hoffman's The Dead Hand is a great, readable overview. If you are specifically interested in the Soviet case, and American efforts to lessen the risk, Graham Allison's Nuclear Terrorism: The Ultimate Preventable Catastrophe has a lot of discussions of both (Allison was involved in some of these efforts, including making sure that the post-Soviet status of Belarus, Ukraine, and Kazakhstan gave all of their nuclear arms to Russia and dismantled their existing nuclear infrastructure).