So far, my summer has been rainy, cloudy and on the cool side with a few hot days. Tomatoes are not growing robustly and not a lot of flowers. Is it a good idea or a bad idea to give my veggies some compost tea? The soil hasn't dried out totally this growing season so I don't know if I'd be making things worse with the tea. The two tomatoes ('Sioux') that are almost ripe are splitting, presumably from too much rain.

I would just add the compost to the soil, and let the rain, worms, and nature break it down.

I already have compost and shredded leaves for mulch. I would probably be using worm compost as my current yard waste compost is full of weed seeds.

I don't see how the compost tea could possibly hurt anything.

Planning on making up a batch this next week. Worm bins need emptying. I figure a mid-season boost is in order as we should be getting some hot weather.

I'm with Tom on this, if water is a problem why add more? compost tea is derived from compost so there honestly is no difference except you are a little less vulnerable to mildews, and fungus when top dressing with the compost vs using the tea. That said compost tea is a great thing to use in the garden but wait until it needs watering.
If overcast, rain and cool weather is a common problem in your area you may want to invest in some red plastic mulch. This will help reflect what sun you do get back on the plant and the red spectrum in light promotes flowering. I wouldn't suggest using plastic mulch on a mass scale but for 1 or 2 rows of tomatoes it should work great for you.

Oh - I don't have rows of tomatoes. They're interplanted with other perennials and such with a couple in pots. The soil is drying out and rain is becoming less of an issue and hot days are starting to take over. I only use fertilizer when I plant and at no other time and am hoping to boost soil nutrients before my short growing season comes to an end.