The non-axiomatic method is common in the work of applied mathematicians. At the turn of the 20th century leading mathematicians like Felix Klein, while acknowledging the importance of axiomatisations, warned that they may be a second fiddle to other fruitful developments where such axiomatisations are not that relevant. Klein engaged a team of top-level scholars to develop a many-volume encyclopedia of applications of mathematics to fields ranging from physics to engineering. Axiomatisations are almost irrelevant for these important mathematical developments.
Felix Klein is one of the superstars of 20th century mathematics and the validity of his work is beyond dispute. Notice that the term "rigor" did not occur in the formulation of your question. Its meaning is dubious and especially at the philosophy SE could come across as naive. Many mathematicians (though by no means all) tend to identify "rigorous mathematics" with "mathematics done in a ZFC axiomatic framework" and from that point of view certainly there couldn't be any rigorous work outside the axiomatic framework, yes. But that's a rather reductive view that would probably not be shared by many editors at the philosophy SE.
Here is a useful post to consult if you think most of mathematical activity has something to do with axiomatic frameworks.