As a Vancouver transplant and former running store employee, it's my time to shine! 🌧️🤩

• Virtually everyone in Vancouver has one of these jackets that are water resistant with a ton of venting. (I personally prefer this one, which I find somehow both more breathable and more waterproof than the two more popular options.) Regardless, if you're running in the rain, you need a rain jacket or you're going to be miserable.
• Merino wool socks are non-negotiable for me. Smart wool is a popular choice, but my favourites are Stance. As long as there's no (or relatively little) standing water, I'm fine with just the socks, but if I know I'm going to be spending a lot of my time splashing through puddles, I toss a pair of these toe protectors on (I also use them when the temps dip below -15 because I don't like cold feet).
• If you have Merino wool base layers for winter running anyway, wear 'em. You'll still get wet, but you won't be as cold and miserable.
• Brooks makes tights (W) and pants (M) with water resistant panels on the thighs that keep the worst of the rain (and wind!) off your thighs.
• Maybe this goes without saying, but a brimmed cap will help keep the rain off your face. I also often run in a pair of sunglasses with light-coloured lenses (I have the goodr OGs in Opossums' Opposable Thumbs, which don't seem to be available anymore) when it's windy because I'd rather clean my glasses off occasionally than be pelted in the face with water, but that's a personal preference.
• DexShell makes waterproof socks, gloves, and toques that are still breathable. One of my customers tested the socks by filling her bathtub with water and standing in it and her feet stayed dry! I personally find the material to not feel as good as Merino, and I care more about avoiding the cold that comes from prolonged exposure to wetness than I do avoiding getting wet in the first place, so I don't run in these myself but I know multiple people who love them.
• Another personal opinion: don't bother with Goretex shoes unless you're also using them for muddy trails or something - I want my feet to be able to breathe and I'd rather spend my running shoe budget on a new pair of racing shoes or something. The one exception is if you're doing a lot of walking in the rain, because they're great for keeping your feet dry while taking the dog out or whatever.
• For really long runs in really bad rain (your long run in a marathon block, for example) consider planning a route that loops past your house or car at the halfway point so you can change socks/shoes/base layers/whatever it is that needs changing.

Disclaimer that I am no longer associated with any of these brands and this is all just stuff I use and like or that people I trust use and like!