Recommendation for Inventory Management Software

Hello, I was recently put in charge of managing my warehouse's inventory and could use some help from someone with more experience. I work for a company that sells Sports Gear both online and in two physical retail location. We also have a partner who has his own Amazon store which carries our brand, but his products also come from our warehouse and this is where the problem arises.

Currently, we don't have any serious inventory tracking system, so many times when our partner sells something, we actually don't have it and he has no way of knowing until an issue has already been created. My boss and him are looking for software that will track the quantity of certain products across two physical/retail locations, one Shopify e-commerce store, and one Amazon store.

Additional issues: the items we are looking to track do not currently have matching SKUs in their online locations. I can fix that with quite the effort but would prefer software that could possibly track one item with multiple SKUs. Next, I do not believe any of or retail products have a proper barcode to scan, so any advice on how I should go about resolving that issue would be very welcome.

Sorry if this is all a lot and/or not completely possible, I was handed this by someone with even less of a clue so I could use all the advice I can get. I'll try to answer any follow-up questions as I'm sure I've omitted some details. Thanks in advance y'all.