Is it necessary to order the adapters and primers for the Illumina libraries from the vendor or the core lab or can I make my own?

Adapters can be home-made and in fact this is the best way to add an internal barcode. Just be sure the phosphate is on the 5' end of the downstream adapter. Email the Core or use your Illumina account to obtain a copy of the latest adapter sequence list. If using custom adapters, remember that acceptable Tm differs by instrument, so you must consider that during the design stage. They also should not interfere with any of the primers in the standard Illumina mixes. It is recommended that you consult with the Core on custom adapter sequences.