On Saturday, The New York Times reported what we probably should have presumed all along: It was Trump Secretary of State Mike Pompeo who "urged" Donald Trump to fire the State Department's inspector general, continuing the widespread purge of government officials responsible for oversight of the impeached president and his team of corrupt incompetents.
The official non-reason Trump gave for firing inspector general Steve Linick was that Trump had "lost confidence" in him, the same catch-all Trump has used to dispense with all other watchdogs who investigated, or merely raised questions about, illegal acts by Trump's team. While the White House seems uninterested in giving any more plausible rationale for the firing than Trump's ever-vocal gut, it does appear Mike Pompeo had a specific reason why he might have wanted his department's watchdog out: Linick had been asked to investigate charges that Pompeo had been corruptly using a State Department employee to run personal errands for himself and his wife.
Mike Pompeo has remained steadfastly loyal to Trump. He was identified as a key player in Trump's withholding of congressionally earmarked military funds to Ukraine in an attempt to force that nation's government into crafting materials to help him smear his presumed election opponent, and defied congressional demands for testimony. He is quite definitely the sort of person who would use government resources to have personal favors done, and would not be the first, second or sixth of Trump's cabinet appointees to be credibly identified as doing so. He is certainly, beyond a shadow of a doubt, the sort to sabotage government investigations into such wrongdoing.
House Democrats are already vowing investigation into Linick's firing; there is no plausible rationale for Trump firing inspectors general across government, immediately after his impeachment, other than as a government-wide attempt to block all remaining oversight into his team's actions. Senate Republicans, as usual, are using Trump's action to either reaffirm their loyalty to Trump over the rule of law or to reaffirm their commitment to saying Trump probably oughtn't break laws while doing not a damn thing in response.
The eternally dumb Sen. Ron Johnson, proven a traitor to his own oath and nation during impeachment, as well as both before and after it, suggested that he was comfortable with the firing because Linick had not been sufficiently helpful in assisting Senate Republicans with an unidentified Senate investigation almost certainly pertaining to continued election-year efforts to smear Trump opponent Joe Biden.
The less dumb Sen. Chuck Grassley, also a traitor to his own oath and nation for the same multi-year patterns of behavior, issued the now bog-standard sternly worded statement noting that Congress requires explicit written reasons for such a firing, and that he will continue to be slightly huffy about that until the precise moment somebody asks him to take an actual action upon which he will fold like an origami swan.
Sen. Mitt Romney, alone in his impeachment opinion that perhaps if top administration officials are doing crimes it would behoove the Senate to at least momentary feign an interest in acting as check against such acts, gave a similar statement. Trump’s actions are a “threat to accountable democracy,” Romney warned, without suggesting he would engage in even the smallest of acts to combat that threat.
So the answer is no. No, it does not appear that the Republican Senate is willing to take any action as Trump continues his purge of those who have been investigating, or who have merely been charged with watching over, his team's continued pattern of grossly unethical and/or criminal acts.
It is likely that Grassley and other Republican concern-bearers will take no actions to support House efforts to call witnesses and probe the reasons for the firings, much less engage in such probes themselves. The party has made it absolutely clear that Trump and his allies are allowed to use government for their personal gain, and are allowed to sabotage any government effort or fire any government employee necessary in order to obtain that gain. They betrayed their country unforgivably in refusing to even conduct a trial or hear from direct witnesses, during impeachment; the play now will be to allow Trump to commit any number of further crimes, rather than conducting oversight between now and November. Trump's corrupt acts will not disappear then, whether or wins or loses, but for Johnson, Grassley and the others, putting off judgment day is paramount. Even if it is only temporary, it is now a party necessity.