When we talk about "vying with" something, we are usually referring to a competition or challenge. To express similar ideas, we can use a variety of synonyms such as competing with, rivaling, contending with, challenging, or battling against. Each of these words has a slightly different connotation, but they all indicate some level of conflict or contest. We might use "competing with" to describe a situation where two people or teams are striving for the same goal, while "rivaling" might imply a more personal or long-term competition. "Contending with" or "challenging" could suggest a struggle against obstacles or difficulties, while "battling against" might indicate a more intense or violent kind of competition.