You can do everything the same day. Other than vinegar, which gets impractical over large areas, pulling the weeds is the alternative. For the weeds that have tap roots, you can use a Weed Hound tool. It allows you to stand up and pluck the weeds out by the root.

Vinegar works best with full sun on a hot day. Vancouver might not have many of those. Last week I tried something different. I had some weeds in full shade. First I wet the leaves with a soapy molasses water spray, then dusted them with baking soda, let it dry, and then sprayed with vinegar. The weeds shriveled up in an hour.

The most effective way to combat "weeds" in turf grass is to provide a good healthy stand of turf and that starts with a soil test so you know what your soils pH and nutrient levels are and then fixing the soil so the grass can grow in thick. Following that is proper mowing and watering. Mow the grass high which will encourage the grass to grow in and crowd out the "weeds" and water as needed to keep the grass growing.
Good soil needs organic matter and these simple soil tests can help guide you in getting there,
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2. Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.
My understanding is that AgCanada does not do soil testing but they may be able to help you find a lab that does.

Who else wagered the standard non-sequitur reply would appear? Pay up!

Nonetheless, the attack plan for a lawn that large is necessarily to use multiple approaches, as dchall implies. Some you'll have to pull. Some you'll spray. Others you'll outcompete. But you want to get the weeds under control, THEN top-dress and seed, THEN work on the reasons why the weeds are taking over. And it will be work. And you should get on it now before dry season comes. And I'd suggest a nice manure or compost tea watering regime.

You can do it, but assiduousness is the key.

Dan

The reason weeds most often appear in lawns in because the turf grass is not healthy enough to outcompete them. A dual pronged approach is most often called for :-) You need to remove the existing weeds AND improve the conditions sufficiently to encourage healthy turf.

Often the weeds themselves will indicate some of the problems - buttercups typically appear when the soil is heavy and moisture retentive. The soil under lawns is typically compacted - from walking or play, routine use of lawn mowers and other equipment and from frequent irrigation. Aeration should be an essential part of your regime to rejuvenate the lawn.

The moss is also an indication of soil issues - compaction, incorrect pH, low fertility, possibly too much shade.

I'd recommend manual removal of as many weeds as possible. Vinegar at this time of year will produce minimal results. Lime to adjust pH to kill off the moss and remove. Dethatch. Aerate with a core aerator. Apply a thin layer (to 1/2") of screened compost, then overseed. You can use a starter fert at this time. Cover with another thin layer of compost and keep moist until full germination (still a bit early for best results in our climate - I'd wait until midApril before reseeding).

When you water this summer, water infrequently but deeply. The deeper the water penetrates, the deeper the grass roots will grow and a healthy root system results in a thicker, lusher lawn that is more able to outcompete weed problems.

The Pacific Northwest tends to be very proactive with ecologically sensitive lawn care and a lot of communities publish various articles outlining the best practices for this area. I've attached a link to one document (rather lengthy but well worth the time spent reading) that should help.

Here is a link that might be useful: PNW - natural lawn care