I'm pretty sure the eruption of Mt Pinatubo in the Phillipines (SE Asia) in the early 90's (June 1991) reduced average global temperatures by about 0.5 degrees celcius (0.9f) - Wikipedia. Given that this wouldn't have affected the main severe weather months in the US (May/June), if you take a very simplistic approach and look at the number of tornadoes in 1992, May saw some pretty low numbers (137) but June picked up with 399. April was also pretty low with just 53 tornadoes. Whether this was, in any way, related to the eruption of Mt Pinatubo, who knows?
Rather than looking at tornado numbers specifically, if you consider the potential impacts of lower surface temperatures arising from an eruptions ipmact on solar radiation, there are a number of possibilities. You could see lower instability levels, however, on the other hand, lower air temperatures could see smaller T/Td spreads which in turn could result in more tornadoes. There are many other possible outcomes. The GoM may not warm up as much and thus air coming up from the gulf could be on the cooler side. So many possibilities. I'd still chase unless the whole of the US was blanketed in a cirrus-like ash cloud.