Collins Dictionary is the only one I found from a quick search that had an appropriate definition:
step up (in American) 1. to go or come near; approach
Oxford only has "step something up".
Cambridge only defines it as taking action or increasing.
Merriam-Webster is unclear if the definition "come forward" is literal.
Collins classifies the definition as American English, and because I am American, I can only confirm that definition and your use of the phrase is correct to me. You are telling someone to move closer, rather than stay where they are.
However, you don't need to say "closer" because that is understood in the meaning of "step up". So you could simply say:
Step up to the car and try to budge it.
Now, if we remove the object from the sentence, I would understand you mean a different definition. So that:
Step up and try to budge it.
Would be understood as this Cambridge definition (also American English):
step up (US) to take action when there is a need or opportunity for it
When understood this way, it sounds like you are setting a challenge. You are telling someone to do it rather than not doing it.
In fact, even the original sentence hints at a challenge being set because of the words, "step up...and try..."