I don't see a problem with a parent facilitating the gift purchases with the child.  I don't think that an ex should necessarily be picking out and buying the present FOR their EX.. unless it is a fairly extreme circumstance.. the child is very young.. the Exes have an amicable relationship etc..

But.. taking the kid shopping.. whether you wait in the car.. or do your own shopping at the same time for other people should definitely be ok.  It is being done for the kid.. not directly as a favor to the other parent.  No different than taking him shopping for a gift for grandma etc...

In fact, maybe it would be a good time to start teaching him a lesson about giving to others?  Dad can help him set up a list of people he needs to buy for.. grandparents.. siblings.. stepparents. parents etc..  

As far as who pays.. does the boy have opportunities to earn money?  Does he get an allowance that would allow for saving some for gifts? Is dad able to afford to bankroll helping the boy buy for his list?  

Maybe once the list is put together.. he and dad can come up with a budget  and given the time before Christmas being short. dad advances him a nominal amount to buy token gifts for everyone on his list.. maybe 100 dollars total.. kid can divvy it up as he sees fit.. and then he will owe dad 10 hours of chores or something like that?  

Dad can certainly drive him to the store... Most 13 yo's are old enough to navigate the shopping experience.. but if he hasn't done that.. dad can be there with him to give him some help. 

Alternately.. Amazon??? dad can let him put a list of buys and dad can go through and vet the list to make sure it is ok?

The extent of your or your husbands involvement should be to drive him to the store so he can pick something out.  He should be the one picking the gift and doing chores to earn the money for said gift.  

Does BM take him to get a gift for dad?

When the boys were younger, DH had them do chores to buy DBDB any Christmas gifts. He would take them to the store and they would pick out what the wanted to buy her and if it was a few dollars more than they had, he would chip in.  Their SF refused to help them because "she isn't my mother" .

Once they had jobs DF stopped doing it and didn't even ask for a ride to the store. 

I helped SS get gifts for BM on occasion, ie, drove him to the store and he picked it out and paid for it. But only if he asked and paid for it himself.

My DH was not willing to do it, but I did it for SS, not for BM. She once sent SS over with a bottle of wine for me, too, for Christmas - must have been in a good mood that day.  I will say, BM never targeted me and this was before the full-fledged alienation took hold. 

"Son, get on Amazon & pick something out that cocos no more than $$, including gift wrap."

No one has to go to the store, wrap or deliver.

At 13, SS is old enough to decide on his own that he wants to get a gift for BM and ask for a ride to the store and/or $$ for a gift. If he's been raised properly, he will do that, and his dad should help him. If he hasn't been raised to be thoughtful and giving,  this would be a good time for his dad to teach that lesson by suggesting that he get a gift for BM. Dad shouldn't select the gift or push the issue.

If SS doesn't have money, dad can give him the opportunity to earn some by doing extra chores. This isn't about BM, it's about SS.

I think there are several missing dynamics here.

Does SS choose gifts for anyone else? Has anyone taught him the meaning behind gift giving? Does SS have to do chores to save up money for anything else? Has SS expressed interest in giving gifts to BM?

Personal opinion is that kids need to be taught whatever the family values are around gifts, and then they need to be held to that standard. If the family is big on giving homemade items, then you teach the kid how to make a meaningful gift. I don't think there should be special circumstances put on kids when gifting to family members, except for practical limits such as price.

If SS isn't required to do chores or save up his birthday money to buy anyone else a gift, then I don't think it's fair to make him do that for his mom. Set a reasonable limit for him to choose something and teach him that it's important to think of others, especially when they think of him enough to get him something he wants. If he wants to get something above and beyond whatever limit your DH sets, then it's fair to ask him to do chores or chip in some of his own cash.

Once he is old enough for a part-time job, however, I think it's entirely reasonable to expect him to pick out gifts for BM and pay for them himself. Freedom of privileges comes with the burden of responsibility. Since his freedom is still pretty limited, so should be his responsibility. That's not to say that DH should pick out, buy, wrap, and give SS the gift to give to BM, but I am saying that DH needs to hold SS's hand through this this year and put more and more responsibility on SS each year so SS just does it on his own.

Like most have mentioned in their comments your SS is old enough to come forth and state his wishes. 
With that being said if he comes forth and asks for assistance it should only be DH assisting him in the process. Due to the nature of your relationship with BM. 
As for a budget it must be discussed prior to going shopping. That way a set price is in mind. 
$100 may not seem like a lot but for a 13 year old that's a lot of chores to work off. 
I would say $50.00 max ! 
 

In our situation we gave all the children a max of $50.00 for their bio parents. They helped pick the items out and had to wrap them. 
 

Good luck hopefully things calm down a bit for you ! 

No way 

Here is MY take on this.

BM can sit and stew in the juices of her own making.

What about facilitating SS13 getting gifts for DH? That has always been my #1. I would always help Munchkin get DH a gift for bday, or Christmas. She got a card on her own with her own money for Dh's bday. She has barely said Happy Birthday to me this year and certainly no presents or cards (and shes a really great artist, and Ive offered to pay for drawings). 

So does SS13 ever get anything for YOU? For DH? Why would BM be more important than you?

I used to do this for my kids when they were little and wanted to get dad an xmas, birthday or fathers day present.  It made them happy to be able to get him a gift. Their happiness and the spirit of giving is what it was about not me getting something for exH.  They had to "pay" through chores and wrap it themselves.  

SS is old enough to chose a gift for BM himself,, take him to a store and wait in the car.  If he cant pay for it himself ,, cap it at $20 to give him and thats it ,,,, after all ,, its the thought that counts ,, $20 is plenty for him to chose a token gift

Nope.  Does SS have allowance? Make him work for the money spent on a gift. NO WAY would I have wanted my DH spending a dime on BM who made our lives hell  even if it is for SS.  Does the bitch buy your DH a gift from SS. If she miraculously does my opinion may change. 

If the Skid doesn't ask, don't bring it up.

If he asks for help, then help.