This would be very easy to do, especially if they make legitimate purchases from time to time. They just have to be the right kind of purchase. Easier still would be to sell services as well as goods.
The right inventory sources:
If they buy things with unclear contents, it's harder to notice that the specific goods they sell are coming from nowhere. The antique store is a good idea. If they buy out old storage lockers, there won't be many people who could demonstrate that the inventory inside should not have included a bunch of specific items.
If they deal in salvage (old shipwrecks, or other things that no longer have "owners" in a legal sense), it's also hard to prove that something they sell is something they didn't find.
The family might have trouble buying enough things like this to sustain the illusion of a high-income shop over a long period of time, but that's another matter.
The right goods:
Some goods will be difficult to track by nature of what they are. Famous paintings are a good example: if a painting is stolen from a museum, owning that painting would be a risky proposition, and so a wealthy art collector with the right contacts might spend a lot of money on the painting but not be in a position to tell other people about it, or how they got it.
If your magical family is... loose... with the law, they could potentially make copies of famous artworks and distribute them as though they were the originals. The buyers would have every reason to help keep the transactions secret (including covering up what they bought), but also would not be in a great position to discover that they'd been tricked into buying one of many copies. The actual business (well, in this case the front) wouldn't even need to engage in much commerce. Selling artwork can generate millions of Euros of income, enough to live on for long periods between sales.
Intangible services are really unclear in what they're "worth", and it can be difficult, or impossible, to prove that someone didn't receive one. If people "buy goods" but are officially charged for the service, you can inflate the cost of the service to make it less clear what inventory you're selling. A pop-culture example of this is
The carwash used by Walter and Skylar White in a later season of Breaking Bad. Their problem was laundering money, but they sold regular car wash services, then recorded them as more expensive types of washes, and it would be very difficult to prove that someone didn't buy a deluxe wash five months ago.
That's a bit harder to mix physical goods with, but they can trade in goods while also hiding their actual inventory of those goods more easily that way. If we go back to the art dealer example from above, they could also provide vaguely defined "consulting" services. The cover story is not so important, as long as the amounts charged are reasonable. If someone buys a "rare" painting for one millions Euros, but it's billed as a professional inventory and art-preservation advising service, the actual sale of the painting won't show up on any records, and it will be hard to prove the "consulting" was worthless or non-existent.
There's no reason the things this family sells need to be mass-produced or generally available goods. A job where people request specific items be made, and those items require hard-to-value inputs like artistic skill or carpentry experience (for example), can do lots of inventory and value tricks.
If they run a mostly normal business, but use magic to save on labor costs and time, they can make great money while still having a totally valid stream of raw inputs. Exactly how this works as a secret approach depends on the limits of the family's powers, but there are tons of ways that they could be more productive than a typical business when creating objects is magical.
How long could it go on?
As long as you want, really, and it gets easier the less money the shop has to make to keep up appearances. But there are easier ways to make a living with magic in the mix than running a secret shop for generations. Much easier would be to use the magic to generate a large amount of wealth, and then simply be a wealthy family indefinitely.
Generating that wealth secretly would have most of the same problems as above, but it wouldn't have to be maintained for anywhere near as long. Once they have the money, and the money is clean enough to use, that's the end of questions.
It is important to note, however, that because they are definitely not engaging in normal commerce they will never be able to have perfect camouflage. It might be very inconvenient to discover, and it might not even be discoverable without someone specifically looking for discrepancies and being lucky at the same time. There will be something abnormal about their business, or else they are just running a normal business, which in this case they specifically are not.