With the usual disclaimers that polling data is notoriously unstable, subject to sampling biases, and easily manipulated by the surveyors if they so desire, there are two datapoints which suggest a majority of Israelis would support military action in Lebanon for the purpose of cowing Hezbollah.
In December of 2023, relatively soon after the October 7th attacks that prompted Israel's current military options in Gaza, 51% of Israelis believed a military strike against Hezbollah to impair their fighting ability was worth the risk of opening a 2nd front to the war. This evidence comes from the Viterbi Center for Public Opinion and Policy Research at the Israel Democracy Institute - the IDI is a non-partisan think tank and seems to have a reasonable reputation for doing good public opinion research. The timing of that poll, however, is important to note. It's relatively proximal to the October attacks and is also much earlier the in war against Hamas. War fatigue can be swift in how it erodes enthusiasm for military action.
The second piece is far more recent (February 2024) poll published by Maariv, an Israeli newspaper with limited circulation and conflicting assessments as to its political leaning. I can't easily locate the poll itself, but it is widely reported, including in reporting from the BBC as showing 71% of Israelis supporting a military operation in Lebanon in response to increased rocket attacks, etc.
The IDI polling is likely of higher quality, but the Maariv polling is more recent which is important when a shooting war is going on.