The answer to the question posed in the title is that while he can fire her as Federal Reserve chair however she would retain her chair on the board of governors.
Fed Governors enjoy the possibility of a 14 year term. It is custom rather than legislation that keeps them from seldom finishing out their terms. The reasons for resigning are obvious, the expertise and insight they gain from an appointment as a Federal Reserve governor is very marketable both in academia and on Wall Street both of which pay more than the Fed position.
Given that there is a moderately fast turnover and Trump is in position to fill the next open Fed governor seats with his own appointees, theoretically he could look to dominate Fed policy in that way.
Only Truman has fired a Fed chair and it is likely that Trump has been advised that firing Yellen could be a political headache that he does not want to deal with, as he already has so much else on his plate.
He will have the opportunity to replace her in February of next year with his own appointee. Whether or not that happens will likely depend on Fed policy from now until February. The results of those policies reflected in the markets will determine her fate.
One of Trump’s longtime confidants is Roger Stone who made cut his political teeth during the Nixon administration, he even has a tattoo of Nixon on his back. Nixon’s method for dealing with the Fed was less direct than Truman’s. He planted fake stories that attacked Arthur Burns the Fed chair at the time. The fake attacks worked and Burns actually came to Nixon for help, unaware at the time that he was the one behind the disparaging stories. After that Nixon had much more say over Fed policy as Burns lived in fear of the attacks returning.
So although Trump may not attack the Fed directly he may employ more “tricky Dick” like tactics to gain greater control and influence over Fed policy in the coming year.
The most experienced Fed chairman in Federal Reserve history is Janet Yellen. That is not hyperbole she is literally serving her fourth term on the Federal Open Market Committee more than any other FOMC participant in history. Her term as Fed chair comes at a time when the Fed has bowed ever so slightly to the pressures from the far right and far left to increase transparency in Fed activities. Things like the “dot chart,” which shows a poll of FOMC members on their forecast for rate increases or decreases and the Federal Reserve post-rate decision live televised press conference are very new and Yellen has navigated them fairly gracefully. And by gracefully I mean she doesn’t create headlines (relatively speaking given her almost omnipotently powerful position) or move the markets much with her appearances.
What may be the Fed’s most powerful weapon public relations wise is their broad network of regional board of directors that are not bankers or academics but most often are respected businessmen or women in the region of the branch they serve. And while before their appointment to their regional Fed board they were likely Republican leaning politically, they have a duty in their position to uphold the institution that they currently serve.
Any manifestation of a showdown between Trump and the Fed will likely be born out of inflation caused by Trump’s infrastructure stimulus plan. This will continue to be a nontraditional and unpredictable presidency. The Fed will fire the first shots in the public debate. Fed board members will critique Trump’s infrastructure plan as overkill and irresponsible given the economic climate.