What is a plant? There are different definitions. By the most exclusive definition “plants” are the land plants including mosses and other organisms that don’t contain hardened tissue for transporting water and nutrients. These are called non-vascular plants. Mosses were first seen around 470 million years ago, long before vascular plants. The most inclusive definition includes single-celled algae that have chloroplasts, giving them the ability to convert light into energy (photosynthesis). It is possible that these chloroplasts formed by incorporation of cyanobacteria, simpler single-celled organisms that photosynthesise. These “plants” are estimated to be as old as 1500 million years, but the older the organism the higher is the range of ages; they could have been as recent as 1300 million years. Cyanobacteria (also known incorrectly as blue-green algae) were first seen about 3 billion years ago. Life existed from about 3.5 billion years.
So you are largely right. It is difficult for life to exist without plants, or at least organisms that photosynthesise. For most of the earth’s history photosynthesis has existed. No animals or fungi are capable of producing their own energy and depend on organisms that do.
How was it possible for life to exist without plants? Photosynthesis is not the only way of producing useable energy, though it is the most important.
Though I find this subject interesting, I was initially surprised that it is asked in a Bible Question website. I am not aware of anywhere in the Bible that discusses this subject. We should not expect God’s Word to be a scientific textbook. However studying God’s Work, including the study of plants, reveals something of the glory of God in his act of creation, which he performed over not just millions, but billions of years. Looking at it that way, the question is a reasonable one even for BibleQ.