Yes, it should be possible to use adaptors and primers from vendors other than NEB with this NEBNext kit. The adaptor must be a T overhang adaptor and the final library yield may be slightly lower than when using the NEBNext adaptors for Illumina®.
Instructions for use with non-NEBNext, full length barcoded adaptors and P5 and P7 universal PCR primers can be found in the manuals for the NEBNext indexed/ UMI adaptor manuals. These manuals can be found at neb.com/E7416 (RNA) or neb.com/E7395 (DNA). Choose the kit manual for the NEBNext kit you will be using. The manuals are written specifically for the NEBNext indexed/ UMI adaptors, the same protocol should work with other adaptors of similar design. We recommend performing a proof of principle experiment with the desired library prep kit and desired indexed adaptors and using expendable samples for optimization of experimental conditions.