The use of lasers to weld polymer sheets provides a means of highly-adaptive and custom additive manufacturing for a wide array of industrial, medical, and end user/consumer applications. This paper provides an open source design for a laser polymer welding system, which can be fabricated with low-cost fused filament fabrication and off-the-shelf mechanical and electrical parts. The system is controlled with free and open source software and firmware. The operation of the machine is validated and the performance of the system is quantified for the mechanical properties (peak load) and weld width of linear low density polyethylene (LLDPE) lap welds manufactured with the system as a function of linear energy density. The results provide incident laser power and machine parameters that enable both dual (two layers) and multilayer (three layers while welding only two sheets) polymer welded systems. The application of these parameter sets provides users of the open source laser polymer welder with the fundamental requirements to produce mechanically stable LLDPE multi-layer welded products, such as heat exchangers.