Yes, it is possible. People who have done excellent tropical work with little Algebraic Geometry background include Federico Ardila, Michael Joswig and Josephine Yu. (I hope I won't insult any of these people by saying that they do not strike me as having much algebraic geometry.)

However, I have had bad luck introducing people to tropical geometry without talking about
valued fields, Grobner degenerations, toric varieties and the other algebraic technology.
I can give a nice colloquium talk or write a nice expository paper where I gloss over this material. But this leaves the reader without an intuition to figure out which questions are reasonable to ask, or any idea of where nontrivial results might come from. This is especially true because so much tropical work right now is not solving specific problems formulated by experts, but in finding the definitions and theorems to make precise the phenomena which people have observed.

**ADDED** I also like Ben's answer. There are parts of tropical geometry which use very serious algebraic geometry, but there are also parts where it is being used for motivation and intuition. You could probably get a lot of what you need from Cox-Little-O'Shea, Fulton's "Algebraic Curves" and some good reference on Grassmannians and hyperplane arrangements.