I need to make barbecue sauce but I don't have mustard, soy sauce, or worsterchire sauce. Is there a way to make BBQ sauce without those ingredients?

barbecue sauces are a fun place to improvise; it's hard to go far wrong. For a general purpose sauce of the type common in the US, you nearly always have sweet, sour(acid), and salt- all of which can be had in a thousand ways, and many insist on some oil of some sort. Smoke is good-the unjustly maligned liquid smoke is great, but smoked paprika, chipotle powder etc. can also help. A lot of people add fruit in some form- juice or jam, usually- which adds some complexity to the flavor, and fits in the acid/sweet profile. If you feel you need garlic, I'd steep some crushed garlic in vinegar- the powdered stuff is just nasty. You can use any spices you want, but keep in mind there may be a certain amount of burning. Personally, I like a much thinner sauce than the commercial versions- I usually base mine in large part on red wine and/or sherry- but that's just me. Above all, enjoy yourself- your friends will still like you if it doesn't come out perfectly.

Don't worry about "start[ing] a knife fight here as barbecue is intensely regional" since you're clearly not trying to pass off your improvised sauce as something that is authentic/traditional to a region. Just call it "My Improvised BBQ Sauce for August 25th" and no one will draw a knife.

I would look at your spices and think about which ones you've enjoyed before on chicken and start from there. Your own taste preferences should guide you what to put into your improvisation.

If you have any citrus (like oranges), you might consider juicing some and reduce the juice to a syrup on the stovetop. Personally I think oranges and chicken go well together.

Also, if you have any jams/jellies, you might consider using some of those. For me, I would favor jams/jellies made from stone fruits (apricot, peach, plum, etc.) for poultry.

You may want to consider a dry rub for your chicken wings as well.

Good luck.

Be careful that you don't start a knife fight here as barbecue is intensely regional. You have most of tlhe basics for Carolina style. You don't need dried mustard and skip the ketchup anyway. Look for some plain yellow mustard like French's. Combine that with vinegar, some honey, and ground black pepper and you'll have a start. Cook over indirect heat because the honey burn giving you a black char that doesn't taste very good. You should taste more mustard and vinegar.
But be warned everyone defends their own turf when it comes to barbecue.

For sure you can make it work without those ingredients. I never use mustard or soy sauce in mine, and you can survive without the worcestershire. Start with the ketchup (though i prefer strained, canned tomatoes instead) . Think of other things that add umami to replace the worcestershir, like some miso paste. Liquid smoke is amazing too if you have it. Molasses and maple syrup for sweetness and a mess of sspices like dried garlic, paprika, mustard powder etc. Here's my recipe and like i said, i'm sure you wont lose too much by omitting the worcestershire.
2cups tomato sauce
1 tbsp worcertershire
1 tsp liquid smoke
1 tbsp smoked paprika
1 tbsp garlic powder
1 onion, diced
1/4 cup apple vinegar
1/4 cup maple syrup
1 tsp mustard powder

Do a search for Alabama in the recipes section here. Maybe one of the white barbeque sauces that come up would work.

I made a bbq sauce with pomegranate molasses that doesn't need any soy and you can omit the mustard. hope this helps! http://bigtastetinyspace.com/2016/04/21/pomegranate-ribs/