Polymer flows through pores, nozzles and other small channels govern both engineered and naturally occurring dynamics in many processes, from 3D printing to oil recovery in the earth's subsurface to a wide variety of biological flows. The crosslinking of polymers can change their material properties dramatically. In engineered flows, polymer crosslinking is often a situation to be avoided. For instance, in 3D printing it is greatly preferred for crosslinking to occur upon impact with a substrate rather than prior to exiting a nozzle. In blood flow, on the other hand, polymer crosslinking can either be advantageous, as in wound healing, or pathological, as in thromboembolism formation. In either of these situations, and in others, it is advantageous to know a priori whether or not crosslinking polymers will lead to clogged channels or cessation of flow. In this study, we investigate the flow of a common biopolymer, alginate, while it undergoes crosslinking by the addition of a crosslinker, calcium, driven through a microfluidic channel at constant flow rate. In addition to quantifying the limits on flowability and clogging in situ in this crosslinking polymer system, we observe a remarkable phenomenon in which the crosslinked polymer intermittently clogs the channel. We observe a pattern of deposition and removal of a crosslinked gel that is simultaneously highly reproducible, long-lasting and controllable by a variety of parameters. We map this behavior as a function of flow rate, polymer concentration, and crosslinker concentration. Our results pave the way for future investigations of the phase diagram controlling intermittent flows in crosslinking polymer systems. Furthermore, the dynamics of the deposition and ablation process suggests an intriguing possibility that the route to clogging failure exhibits signatures of chaotic dynamics.