Can I descale/decalcify my coffee machine without a special product?

The manufacturers will tell you:

During decalcification with decalcifiers on citric acid basis, residue can form and clog the fine lines and valves of your espresso machine. Lime deposits can become sealed under an unremovable layer of residue formed by the citric acid.

A similar reason goes for vinegar. And Citric acid (from what I'm told) is less problematic than vinegar.

The real expensive stuff like Durgol isn't actually citric acid and some other formula (unlike citric acid it tastes awful).

Here is their description (pdf)

While your machine's specific kit might be dosed for perfection, you're likely ok with any descaler for the size of the machine. A competition grade professional machine on the other hand, I'd recommend vicious service. If you only see what's under the shower screen of some of the machines out there in the wild.

A key point is that they claim their product is faster. So if you have time, and are willing to rinse a quite a few more times, I wouldn't worry too much about it.

The Cafiza powder from Urnex is really good at cleaning coffee grime, but does not de-scale/de-calcify. I would recommend it for cleaning coffee urns and the coffee stained parts of your machine.

Finally, you might wish to use softened water. It'll make the coffee taste better and reduce the build up.

I did with citric based solution and it worked for while but then the machine started leaking water from inside. It works but it can degrade the hoses and valves inside the machine. It is not worth the potential damage to a $750 and up machine. Just spend the money to buy the right product for descaling. Your machine will last longer. I learned the hard way.

Descaler products for coffee machines contain sulfamic acid which is stronger than citric or acetic acid, plus corrosion inhibitors which prevent it from attacking metals. You don't have to buy a descaler from a coffee machine brand (which will indeed be more expensive), but there are generic products with sulfamic acid out there. One important point - sulfamic acid slowly hydrolyzes in water, so liquid products have limited shelf life. Prefer tabs if you want to buy in bulk.

You can use vinegar or citric acid if you don't mind a less effective product. Here, the number 1 mistake people make is not waiting long enough. You need to wait roughly 5 times longer with vinegar, compared to a descaler solution. If your coffee machine need to be descaled for 10 minutes, you'll need almost an hour with vinegar. This may be difficult to achieve if your coffee machine has an actual descaling program, you'd better buy a suitable descaler in this case.