On Monday and Thursday after Hagba on days that Tachnun is said, the Chazan says aloud a few Yehi Ratzon's immediately after Hagba was done while Gelila is still being done. Suppose the Chazan had Hagba what should be done? Should someone else say the Yehi Ratzon's? Should the Chazan say it after Gelila is completed? Should the Chazan say it while seated and the Golel is doing Gelila? (sources please)
The answer here, on the bottom of p. 3 may surprise you, as it surprised me.
According to Rav Amram Ga'on, the Golel is the one that should be reciting this prayer all the time, not the shat"z. Note that he says "The last one rolls the Torah", meaning that the person who gotthe last Aliyah is the Golel. Also, note that he doesn't mention a magbi'ah. It's possible that he was following what I have seen now used in Sefardi nuscha'ot where Hagbah is done first before reading begins.
Having the golel recite Yehi Ratzon not common practice, but, as I haven't yet seen a halacha that addresses your question directly, I wouldn't be surprised that someone states to follow Rav Amram's ruling.