Yes, tomato seeds will germinate without a year having to pass first (even right away, before you even dry them). Just make sure it's not too hot or cold for them to germinate and grow well. If it's consistently very warm in your house, they might have trouble, especially if you have artificial lights on top of that (since artificial lights can produce heat; even the low-heat ones, like CFLs, will make the room more noticeably warm when it's naturally kind of warm).
Edit: However, it should be noted that tomatoes germinate faster in warm conditions (as long as it isn't too warm) than they do in cool ones, in my experience.
Anyway, I've grown newly harvested tomato and pepper seeds before with good results. They don't need to be aged. New is good.
As long as your growing season is long enough and your tomatoes tolerate the temperatures they will be in (indoors and outdoors), you should be fine.
Edit: If you want the plants to fruit indoors (rather than start them indoors and transplant them outdoors), as Stormy seemed to me to understand, then be sure to give your plants enough red light (which helps with flowering; blue light helps with leaf growth), soil, and enough of the proper nutrients. I was assuming that you were wanting to start them indoors while it's still summer outside, and then transplant them outside for the remainder of the season, hoping to get some extra tomatoes during the rest of this season. Whatever season it is, indoors or out, you'll probably want your nighttime temperatures to be between about 55° F. and 70° F. Many people keep their house above 70° F. at night; so, that's something to watch out for. Daytime temperatures should be between about 60–85° F. Flowers might drop if it's too hot.
If you have heat and cold tolerant tomatoes, parthenocarpic tomatoes or such, then your plants may set fruit in a wider variety of temperatures. See this question/answer for information about such varieties.
I concur with stormy about low nitrogen fertilizer for indoor use, but if your soil outdoors is already low in nitrogen (like mine probably is), then adjust accordingly. However, for plant growth, I haven't had issues using an even fertilizer (20-20-20), even though slightly less nitrogen than potassium is generally recommended for tomatoes. However, be careful, anyway, I've found that my indoor tomatoes can get burned by 18-18-21 fertilizer (but 20-20-20 fertilizer doesn't burn at all), which doesn't make a lot of sense, unless with the 18-18-21 fertilizer they're just using a faster-release nitrogen, or the potassium didn't dissolve in the water properly or something. That's one reason I like to mix fertilizer proportions myself (I know it really is what I think it is, and I can adjust it if it isn't what the plants need.)
However, as far as fluorescent lights go, be careful there. A lot of people will tell you to get only 5000-7000k bulbs, which is great for growing seedlings into plants, since it's high in the right kinds of blue light—but lower color temperatures are actually higher in red light, which is important for flowering. Lower color temperatures also help protect against damping off disease, in my experience. Plants don't lean toward the lower color temperatures as quickly, though (but when they adjust to it for a while, they do lean toward it faster than they initially did). They can still grow leaves on lower color temperatures of fluorescent bulbs (I know from experience), although 6500k might be more ideal for that.
