(The converse is easy.)
is there an elementary proof that if two angle bisectors in a triangle are congruent, then the triangle is isosceles?
It depends on what you mean by "elementary." There's a quick proof if you use the two facts that (1) the angles bisectors of a triangle intersect at the incenter, the center of the inscribed circle, and (2) tangents drawn from a point external to a circle are congruent.
(To get you started: Let the triangle be ABC with the angle bisector of A hitting BC at D and the angle bisector of B hitting AC at E. You're assuming AD=BE and you want to show that AC=BC, or equivalently, angle A is congruent to angle B. Draw the inscribed circle, and look at it from the perspective of point C...)