The word 'sushi' actually refers to the property of 'being sour' and refers to the way the short-grain white rice used in sushi is treated with rice-wine vinegar to give it that sticky, sour and slightly sweet taste. Finding the perfect balance between rice, water and vinegar can be the key to making successful sushi, however it is often difficult to source some of the ingredients, such as the rice-wine vinegar, that is called for in the traditional recipes.
Will different vinegar affect the taste?
To answer your question briefly, replacing the type of vinegar used will change the flavor. How noticeable that is, and more importantly- how it actually tastes- will depend on your palate, the type of vinegar you use, how much of it you put in, and whether you can balance the flavor with the addition of another ingredient. The ratio that I find works well for me is 3 cups of sushi-rice, to 3.3 cups of water, to 1 cup of vinegar. Tweaking this basic ratio usually results in tasty rice that sticks well to the toasted seaweed or nori sheets when rolling sushi.
What could I try using instead?
If you are desperate to make sushi but just don't have the right vinegar, you could try using normal grape-based vinegar or apple-cider vinegar and then perhaps adding a pinch or two of sugar and see if that balances the lack of sweetness or even adds an extra dimension to the flavor of your sushi.
In fact, often times finding yourself short of an ingredient can be a good thing as I once found when I ran out of rice-vinegar when preparing some sushi; I decided to use a 'red apple' vinegar which gave the rice a pink-tinted hue that seemed to really impress everyone that tried it!