Can stock exchange workers go on strike?
The Stock Exchange workers in US could strike if they wanted. The right to strike in US is enshrined in the US National Labor Relations Act of 1935. See also explanation of the law by the National Labor Relations Board  to see conditions under which strike can occur. None of the conditions seem to exclude stock exchange workers.

However, as @BrianRomanchuk points in his +1 comment, strikes are normally called by unions. According to the statistics published by U.S. Bureau of Labor Statistics in 2019 by occupation only $4.2\%$ workers were members of union and only $5.1\%$ represented by unions in the "Business and financial operations occupations" and if we look more narrowly only at finance as industry (which is arguably more appropriate for this question) the figures become $1.4\%$ and $2\%$. This being said I could not find any statistics specifically for only stock exchanges, but I would expect that unionization rates there would be lower than the aggregate for whole industry.

At the same time statistics show that people in the US working in finance and insurance industry were in 2018 ranked 4th in terms of average salary (see here). 

So to sum up, people in US working in financial industry are not generally members of unions or even represented by unions. This would presumably make organizing any meaningful strike difficult.  At the same time people working in finance industry in US are already in one of the highest paying professions meaning that there is most likely not much incentive to strike.