Yes, it is possible to sell t-shirts through Instagram direct messages (DMs). I'd recommend using an online payment processor such as Stripe (https://stripe.com/), so you can accept orders and track your revenue. As your business grows, consider using an e-commerce platform like Shopify (https://shopify.com), so you can reach more people.
You could, however, it is likely you will never be able to expand your business beyond family and friends. It does not cost that much to set up a payment method on an actual website. I would suggest starting your business on Etsy, as they basically give you a website and payment method for free. Obviously they take a cut, but then you can have your Instagram link to the etsy account and basically have that be your business until you are ready to invest in your own website!
There is a way through a method called comment selling. There are two companies in this space. One is called Soldsie and the other is called Comment Sold. This triggers an invoice to them via email when they comment on your t-shirt post. They say "sold" in the comment and get an invoice delivered to their inbox. It's really cool. It's not via DM, but it's another way to sell on social media without having an eCommerce site.
Its not impossible, and unlike all other answers here, it could even be a exclusivity based concept…. I totally believe that if you setup a plausible strategy, this could be a innovative way to create a exclusive brand. Do understand, that you might at some point reach a limit on your own resources, so you might need to hire new collegues to do the same.... I mean its unusual.. but hey sometimes unusual is what sells
Of course you COULD, but it's not likely unless you can overcome a number of people's hesitations.
First, the question of "is this person/store reputable or a scam?"
It's difficult to prove you are a real business without a website or something to establish trust with the prospective buyer. You'll need testimonials, a substantial following, or other forms of social proof.
Next, you have to make sure people feel comfortable sending you money. What is the guarantee that you won't just keep it? Once money comes out of your Cash app or Venmo, it's not coming back. When I order something from Amazon, I know that I can talk to a customer service rep in the event that ANYTHING goes wrong. What assurances are you giving people on Instagram that you can offer a comparable experience?
In the end, you're probably making things more difficult for yourself. Use Instagram to promote your products, use the DMs to drive them to a payment portal or store, especially if it is one that is familiar and trustworthy. Consider using any of the existing custom t-shirt and product selling platforms out there.
Let me know if you'd like to discuss this further on Clarity.
If I'm understanding your question correctly, it sounds like you want to bypass using a payment merchant like Paypal or even Etsy, Shopify, etc? If you want to make money, you will need a way to collect the money from people, unless you will be doing a cash transaction in person. You can use venmo or Square to accept online payments without setting up an online store. You can try apps like Depop.
It is possible to sell T-shirts successfully without using a payment method. However you will not be able to expand and it must be done within your city. So you can put where you’re based on your instagram’s page. The payments can be done in person. You can do pick-ups at your own space or also do the deliveries and charge an additional fee for that.