The spokeswoman for Judicial Watch said Bryan Pagliano, a Clinton IT aide, declined to answer questions when subpoenaed by the group. Pagliano was involved in setting up and maintaining Clinton's private email server and accepted an immunity deal with the FBI earlier this year in which he cooperated with their investigation into the server.
Pagliano's attorneys have said their client is "caught up in a lawsuit with an undisputed political agenda" and that he would not cooperate with Judicial Watch.
The Republican National Committee pounced on the news, reported earlier in the day by Fox News.
"The fact Hillary Clinton's former IT staffer pleaded the Fifth an astounding 125 times is another reminder of how much she has to hide and how serious the FBI's criminal investigation really is," chairman Reince Priebus said in a statement.
Pagliano, who was hired by the State Department after a stint as IT director for Clinton's 2008 presidential campaign, and was paid separately by the Clintons to perform work on the server, located at their home in Chappaqua, New York.
Emails released to Judicial Watch Wednesday paint a fuller picture of the challenges presented by Clinton's email server.
In one chain from early 2011, Justin Cooper -- a longtime staffer for former President Bill Clinton -- exchanges emails with Clinton aide Huma Abedin after Abedin and Clinton experience trouble accessing emails.
Cooper tells Abedin the server was taken offline after a hacking attempt was registered.
There's no indication in the emails that the server was breached, and Clinton's campaign has long maintained it was never compromised.
In a separate chain from 2010, Abedin expresses frustration over the fact that emails from the server aren't reaching people on the State Department's email server.
"Technicians implemented a series of troubleshooting measures to the department's system -- not Secretary Clinton's system -- to attempt to remedy" technical issues reported by employees involving email spam, according to State Department spokesman John Kirby.
But the issue caused Abedin to suggest to Clinton: "We should talk about putting you on state email or releasing your email address to the department so you're not going to spam."
These exchanges were first brought to light in a State Department Inspector General's report released last month.