No, you may not do this legally without permission in the form of a license from the owners of this intellectual property.
Your video game based merch business plan is a horrible, horrible idea.
There is no reasonable way that you could have known just how horribly awful and bad an idea this was without talking to someone familiar with the law. So, I'm not saying that this was a stupid or unreasonable question. But, now, you know. And, you should run away from this idea as fast as you can.
What you are proposing to do is blatant infringement of copyright (and trademarks) through the creation of derviative works, on a systematic basis, for profit, without permission, in a manner that does not constitute a parody or satire or any form of fair use.
This kind of economic activity is precisely what copyright and trademark laws are designed to prevent. The case against you for liability could only get more clear if you were selling pirated copies of the game itself. You would have no legal defenses (other than statute of limitations if they waited to many years to sue you, which they almost certainly would not).
You would be liable for statutory damages of up to many thousands of dollars per infringing item and the attorneys' fees the intellectual property owners incurred to sue you. The owners of the games could probably get a court order to destroy all of your merchandise, and a restraining order and injunction to force you to immediately shut down your business at any time. They could obtain all of those remedies without sending you a cease and desist letter before suing you.
The moderately likely worst case scenario economic liability that you would face would be on the order of 100 times the amount of profits you could hope to make in a best case scenario, and the likelihood that you would incur some significant civil liability is on the order of 85%-90% (with almost of of the little or no civil liability percentage attributed to scenarios in which the company doesn't notice that you are infringing upon its intellectual property rights). Also, the more profitable you are and the higher the volume of goods you sell, the more likely you are to be sued. The liability risk to profit ratio grows with each additional dollar of profit you make.
There is a good chance (perhaps 65%-75%) that they could establish that your violation was willful and wanton in these circumstances, which would also prevent you from discharging any part of the massive judgment against you in bankruptcy.
So, there is a better than 50% chance that you'd be stuck with an intellectual property rights infringement debt, which could easily run into the high hundreds of thousand or even many millions of dollars, plus post-judgment interest at a rate similar to the market rate for high risk junk bonds, for the rest of your life.
In terms of the economic harm involved to you, this would be almost as bad as having all of your property seized and then being sold into slavery for the rest of your life, if a coin toss bet comes up tails, but you get to keep the coin if it comes up heads.
A settlement in which you turned over every penny you ever made in the venture, destroyed all of your products, shut down the business and paid them an additional low five figure amount in lieu of penalties and attorneys' fees would be a very generous offer.
You would also face a real risk (perhaps 10%-15%) of some low level felony criminal liability (perhaps several years in prison).
This is a business plan that is so toxic with immense, near certain liability risks of the worst possible kind that you should put on gloves before picking it up and tossing it into your nearest available fireplace or campfire.
This business plan poses more liability risk to you than opening up a nightclub, bribing your contractors and code inspectors to ignore all fire and electrical codes, painting the walls with turpentine, padlocking all of the exit doors from the outside, changing a $1 cover and selling booze at cost to get huge crowds, and then booking bands with lots of fireworks in their stage shows on a nightly basis. (Yes, I really had a client who was stupid enough to come up with this business plan until I talked him out of it.)
From a civil liability perspective, you would have less exposure to economic liability if you started a business that involved abducting random cats and dogs and goats off the street and charging customers to forcibly rape and then mutilate them, while filming it for distribution on the Internet with your real name and fingerprints in a watermark on every image and close ups of the animals collar tags and the goats' brands. The likelihood of criminal liability would be quite a bit greater (perhaps 60%-80%), however, even though the punishment if you were convicted of felonies for the bestiality business would be similar to the punishment for a conviction for copyright and trademark infringement. (Thankfully, I have yet to see a client try to implement this business plan.)
Go hire someone to secure a license from the intellectual property owner for you (perhaps an IP lawyer or an agent or broker), or forget about it. This is an industry where you absolutely must be legitimate, and you have to go big or go home. The economies of scale are simply too immense to ignore. If your anticipated gross sales aren't at least $500,000 a year or so, you probably shouldn't even consider doing it whether it is legal or not.
A license, if you could get it, would probably cost $6,000 to $12,000 in professional fees to negotiate (if you could accomplish this feat at all), perhaps a similar amount for an upfront fee to the intellectual property holder, and probably 10%-35% of your profits on an ongoing basis. This is because the market rate of licenses of this kind are geared towards what large scale distributors selling wholesale to Wal-Marts, department stores, and national mall chains could bear (I have some clients that happily pay these kinds of license fees so they can sell branded products, manufactured in China and then imported, on a high volume national basis.)
The license fees would be a huge bargain by comparison to your economic exposure to infringement liability. But, if even the licensing fees (if you could negotiate a deal to pay them) are too expensive, then, this business plan doesn't make economic sense even with permission in the form of a license from the intellectual property owner. In that case, you should instead go into the lemonade stand business or buy a food truck, or become an Uber driver, or open a coffee shop or a liquor store, or something like that, with less liability risk and a more proven business model.
If you must make video game merch without permission, do it in a back alley of a small town in rural Mexico on a cash only basis (where lots of people do stuff like this without getting caught), rather than an online store, where any bored paralegal or network manager in the video game company's law firm can find you at a moment's notice and might win a promotion or a raise or a bonus for doing so.