The reason we do not find requests in the Rosh HaShanah prayers is because the Zohar says that we do not ask for material things on Rosh HaShanah. The thrust of the Rosh HaShanah prayers is to attain a closer relationship with God. We ask that God should place His awe upon all His works. We ask that He rule over us directly instead of through intermediaries. We want to come close to Him. On Rosh HaShanah we proclaim His kingship over all. This, rather than personal requests is the theme of the Rosh HaShanah prayers.
Since requests for material things are purposely omitted from the Rosh HaShanah prayers, why is it permitted to ask in the form of a symbol? The Sfas Emes explains. The reason God created things with symbolic meaning is in order to use them in our service to Him. The Zohar calls this "remiza dechachmesa/hint of wisdom." We can look at every thing and find in it a hint, a symbol, that points to serving God. In fact, the symbolic meaning could well be the main reason for their existence. Using symbols to ask for material things points to this concept. To the extent we want the material things to help us to better serve God, we are permitted to ask for our material needs through the symbolism of the foods. After all, the very existence of these foods is intertwined with what they symbolize.
On Rosh HaShanah, God gives the Creation new life for another year. How can we partake of the blessing that God bestows upon the Creation? The key lies in our intention when we accept the life (and all the things) that God gives us. We receive according to our intention to use what we are given to serve Him. May we merit intending to use everything God gives us to better serve Him!