Can I add a neutral bus bar to my panel to allow more connections?
You CANNOT add a neutral bus. You can add a ground bus.
Neutral is not ground.
The essential difference here is that neutrals handle normal service current 24x7... whereas grounds only handle current during a ground-fault event.  So the neutral bars have thermal considerations ground bars do not, as well as, you don't want current to normally be present on the panel chassis, which is grounded.
So you cannot add additional neutral bars, but they provided enough neutral slots for your needs, so you are all set.
You can either add additional ground bars, or use the existing spaces as effectively as you are allowed to.  See below.
UL would never have approved what you claim.
Are you sure you have a 40-position panel?  If you have a 20-space panel, that does not mean you have 40 positions; some spaces are surely forbidden for tandem/double-stuff breakers if they only gave you 40 neutrals. You must heed those restrictions and not cheat around them.
In order to be approved by UL, a panel designed for n circuits MUST provide n neutral bar screw lugs, and also n / 3 (rounding up) ground screw lugs.  That is because NEC and UL both require each neutral be entirely solo on its screw... but up to 3 grounds can share 1 screw provided the panel labeling allows this, which it usually does.
Typically, on cheapie panels, they provide only neutral bars, and simply provide extra screws (1/3) on the neutral bars for the grounds, on the assumption this is a main panel.  If it's a subpanel, you are expected to purchase accessory ground bars ($4-7).
I do not believe for a New York minute that a 40-circuit panel shipped with 40 neutral screws only and no grounds.
That leaves only 2 possibilities:  The original installer squandered the screws by wiring many grounds solo... and you'll need to triple up the grounds (or otherwise follow label instructions) so that have enough screws to solo the neutrals.
Or, the panel has been illegally double-stuffed (e.g. using non-CTL breakers in a CTL panel) and it has more circuits in it than it can support.  That will need to be corrected, may be time to think about a subpanel.
Read the instructions carefully on accessory ground bars
Some of them do not allow 3 grounds per screw terminal, some only allow 2 or even 1.
However you must &quot;stay within brand&quot; of the ground bars; the panel labeling will specify specific models of ground bar, and they will bolt right up to the sites already drilled and notched for those bars.  Easy peasy.  You could use a 3rd party bar with AHJ approval, but it'd be an awful lot of work to drill and tap custom mounting holes for it. Not worth it for the $1-2 you'd save at most.