What is the alternative to a vine ripened tomato?

Tomatoes grow on a vine. But it is possible to pick them unripe, ship them unripe (which is much easier than shipping ripe tomatoes), then gas them with ethylene at the destination. Ethylene acts as a plant hormone and causes ripening.

But tomatoes ripened in storage don't taste the same as vine ripened ones. The compounds a tomato builds are dependent on the amount of UV light it gets, the surrounding temperature, the speed of ripening, the nutrients it gets during ripening, and many other variables. There are lots of tasty compounds it creates while ripening on the vine, which are absent when it ripens in storage. This is why vine ripened tomatoes are tastier. It is also more expensive to let them ripen naturally, therefore the producers label the vine ripened ones as a sign of quality, else customers wouldn't be willing to pay the premium.

Many fruits (tomatoes being one) and vegetables are picked before they are ripe and then artificially ripened at their transport destination using artificial means like ethylene gas. This makes fruits and vegetables make it to the store and last longer there without spoiling, and is the reason we have many of our vegetables year-round. The down-side to this technique is that the flavor of them is nowhere near what a garden-ripened vegetable or fruit would be.

A vine ripened tomato is one that has been allowed to ripen completely on the vine until it is at or near its peak, giving much better flavor. Because they are so perishable they tend to be much more expensive and harder to get. Supermarkets won't have them but farmers markets will depending on the season. Good restaurants pay a lot of money for a good year-round supply of quality vegetables.

I grow my own and I can tell you that there's not comparison between even the best store-bought tomatoes and the ones from a good garden. Not hard to grow either if you pick the right variety.

That being said, the term vine-ripened is often misused as a bit of marketing speak to make them sound more attractive, however if you're at a good restaurant it's probably true.

To contradict / complement the other answers: the "vine-ripened" point is mostly moot. Your regular old comercial round red tomato variety has been bred for shelf-life for roughly forever, won't develop much flavour anyway, and most of whatever is left is probably lost in (refrigerated) transportation and storage even if they were picked ripe - I haven't found one that wouldn't taste disappointingly for years.

What you're looking for is a tomato that's sweet and tasty, so go with the one you can get that you like best. Heirloom/named varieties are a good choice. (E.g. Roma or San Marzano - those are denser-fleshed canning tomatoes but should work fine in a sandwich, especially if you don't like juice making things mushy.) The kumato is also tasty even before it fully ripens. Buying local from a market helps. Avoid refrigeration, eat them ASAP.

In many countries, including the UK, tomatoes grown outdoors often will not ripen at all, due to a lack of sunshine at the right time. What gardeners usually do is pick the green or reddening tomatoes and leave them to ripen on a windowsill (or something similar). They may not taste as good as ones grown under glass. but they are better than ordinary 'vine-ripened' tomatoes would be. [Feel free to convert to a comment; I don't have an account here].