How to get this electric panel up to code?

Guess you'll have to go shopping for secondhand electrical parts

The good news is that with all the labeling in hand, it's rather easy to figure out which cover you need; the labels say right on them that this takes a HOMC20L125 cover. However, your loadcenter is of an older design (Series 03) that has been discontinued for some time now, and according to this Square-D FAQ, the HOMC20L125 cover is no longer available from Square-D, with no substitute parts provided. So, you'll have to find either the cover itself or a complete HOM20L125C series 03 loadcenter (note that Series 01 and 02 use a different cover P/N, and so do the series S0x Homeline loadcenters as they have convertible mains, a feature your loadcenter likely lacks given the lack of height up top) that you can harvest the cover from.

In addition to replacing the cover, it seems unlikely that the neutral bus bars allow two wires per screw (especially considering there are 40 screw spots in a 20-space panel, plus the added accessory ground bar). Permitting two (or more) ground wires per screw is common, more than one neutral wire (or mixing ground and neutral is not). The neutrals should likely be redistributed so that there is only one under each screw.