You are correct. The researchers here sequenced each vaccine starting with the bit of vaccine left in the vial after administration. The goal was to get a raw sequence of the Moderna mRNA component so it can be easily filtered out as being a signal of therapeutic origin. Pfizer's sequence has already been published; it's incldued here to confirm that the result achieved experimentally matches the published sequence.

The authors reverse engineered the sequences of the vaccines, obtaining them from the remaining mRNA present in the vials.

“Assembly” in this case means that they merged several short sequences they obtained, each representing a fragment of the whole mRNA sequence.

They sequenced vaccine leftover remaining in used vials.

So reverse engineering basically.

And reverse engineering only sounds dramatic until you take a step back and acknowledge that it's what they literally do all the time. Only that usually the sequences they read are not the outcome of some human development effort but of naturally occurring evolutionary processes.