I know that in general it is forbidden for one to pray to God for his physical needs on Shabbos, and hence we don’t directly pray for the sick on Shabbos, and don’t pray for our livelihood. What about this year that Rosh Hashanah falls on Shabbos, does this prohibition still apply or do we say that since it is judgment day therefore one may pray for his physical needs even when it falls on Shabbos?
Yes. One may make requests of physical and spiritual matters on Rosh Hashanah even if it coincides with Shabbos. The requests are to be made after the first Yehiyu Leratzon but prior to reciting the second Yehiyu Leratzon.
Explanation: Although asking for one’s physical needs is ordinarily forbidden on Shabbos and Yom Tov, nevertheless the Poskim conclude that it is permitted to do so on Rosh Hashanah being that this day was established to be a day of judgment which requires prayer and supplication.
Sources: See regarding the allowance to pray for physical needs on Rosh Hashanah even if it falls on Shabbos: Mateh Efraim 584:25 [regarding an ill person]; Az Nidbaru 14:21; Piskeiy Teshuvos 582:7 Vetzaruch Iyun from the ruling in 584:5 regarding Avinu Malkeinu, that it is omitted on Rosh Hashanah due to Bakashas Tzerachav! Some suggest that only a set prayer of request was negated on Shabbos Rosh Hashanah while a private request is not only allowed but is motivated to be done. [Piskeiy Teshuvos 582 footnote 21]; See regarding the prohibition to ask for one’s physical needs on Shabbos and Yom Tov: See Admur 294:1; 299:15; 188:4; 584:5; Where in Shemoneh Esrei are the requests to be made: See Admur 122:1; M”B 12:3