We use Lincoln and its easy to get C of C's.  We also use ESAB, and I have to get them from my salesman, but again we buy so much I have it within 2 hrs.
Before you buy the electrode, make sure that you can get one or don't buy that product, pick another manufacturer.  Especially don't go to Harbor Freight and expect them to give you any type of paperwork on their electrodes.

When dealing with traceability issues, a good auditor will not accept anything less than a melt certificate from the manufacturer, traceable to the purchasing document and to the shop order where the work was performed, showing that heat number (aka melt no., lot no., etc) on each document in turn. If one of your welds fails and you cannot show this traceability, you are very likely going to be liable for the consequences of that failure, especially in aerospace.  I can only guess at some of the difficulties you have been experiencing; flag-tagging, traceability, availability of smaller quantities, proper unit packaging and identification to name a few.  You will find, as did I, that you have to reach out for suppliers.  Every order I make is out of state.  I can PM you with more info, if you want it.

I have never understood the requirement for a Certificate of Conformance...

When Lincoln or whoever builds a batch of elecrode, puts a label on it that states it is E71T-1M, and puts it on the market as a E71T-1M, why would there be a question of whether it in fact meets the requirements of E71T-1M?

I mean if it is thought that the manufacturer may falsely claim the product is E71T-1M, why would we then believe their C of Cs are somehow going to be legitimate?

If traceability is a job requirement, you need to request the material test report when you order the filler metal, not after delivery.

You can get whatever is needed if you know how to specify the material properly. AWS A5.01 (ASME SFA5.10) is a procurement guide that tells the purchaser how to properly specify the filler metal and what documentation is "deliverable."

There are several "schedules" defined by AWS A5.01 and used by the purchaser when issuing the purchase order. In cases where it is necessary to have a material test report for each "heat", "lot", or "batch", the purchaser will find it best to speak directly to the manufacturer. You can get material test reports that are traceable to each container of filler metal if you specify the proper “schedule” in your purchase order.

Filler metals that are required to have traceability are usually available directly from the manufacturer. The manufacturer typically "qualifies" the filler metal once a year. The heat and/or lot tested serves as evidence the quality control system in place is functioning properly. The heat or lot that is tested is often placed in the "bonded warehouse" for those customers that require full traceabiliity. The standard "off the shelf" filler metal is produced following the manufacturer's standard quality control system and the label slapped on the container and shipped to their distributers.

Under both the AWS and ASME filler metal specifications, the label affixed to the container of filler metal is the "certification" it meets the requirements of the applicable filler metal specification. Anything beyond that label must be specified at the time the filler metal is ordered. Having all the requirements properly spelled out in the purchase order is the secret. Use AWS A5.01 as your bible and you will get what you are looking for.

One last thing I like to include in purchase orders for filler metals purchased from local suppliers, "no repackaging" by the vendor is permitted. All filler metals must be delivered in the original unopened manufacturer's packaging. 

Best regards - Al