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The best you're going to do at $500 (but still see significantly improved picture quality) is probably a $200-ish consumer grade camera and the rest on a capture device. for HDMI to USB3, I've used this one many times and it works well:

https://www.amazon.com/Magewell-XI100DUSB-Video-Capture-Dongle/dp/B00I16VQOY

But that doesn't leave much room to play for a camera and stay around $500. For consumer cameras I like the picture quality of the Canon Vixia line, but Sony, Panasonic, etc. have similar consumer cameras at similar prices. Here's one that fits the budget:

https://www.amazon.com/Canon-VIXIA-R700-Camcorder-Black/dp/B019UDIL80/

This one even has a mic jack, which is a nice plus.

You might be able to save even more on the capture device by getting something like one of the Blackmagic decklink capture cards, if you have a desktop to put it in. Still, you're only going to save $15-$25 or so.

If you can put your budget into the $1,500-$2,000 then you could move up to a true prosumer camera or even a professional camera if you buy used. A used Panasonic HMC-150 should be going for about $1-$1.5k on ebay. You'd really see a jump up to pro looking video at that point.

HD webcams that are connected to the laptop just with regular usb?

Yes or via active usb extender cables such as
https://www.amazon.co.uk/BlueRigger-Active-Extension-Repeater-Female/dp/B005LJKEXS/ref=pd_lpo_147_bs_img_2?_encoding=UTF8&psc=1&refRID=5XTF7EG145X9N1PZY2Z9

Any suggestion to which one? Could you suggest mics and mixer?
i use a cheap behringer usb mixer
https://www.amazon.co.uk/gp/aw/d/B008O516L0/ref=mp_s_a_1_6?ie=UTF8&qid=1481858123&sr=1-6&pi=AC_SX236_SY340_FMwebp_QL65&keywords=behringer+usb+mixer&dpPl=1&dpID=61gv6f5oIkL&ref=plSrch

many good quality mics available for under £100. you could even just use the mic on one of the USB cameras and do away with the mixer altogether if it was close enough to the person - audio quality will not be as good as using dedicated mics. if you let me know about the anticipated set up / scene you're wanting to broadcast i can advise in more detail

I actually have a VIXIA R500--I'd forgotten it has HDMI out! So really, all I need is the capture card. I've got a desktop I can put it in, so it looks like an internal one is the way to go. Is there any reason not to go with this one?

https://www.amazon.com/Blackmagic-Design-DeckLink-Recorder-Capture/dp/B00CDIXCSO/ref=sr_1_1?ie=UTF8&qid=1479746927&sr=8-1&keywords=Blackmagic+decklink

Thanks again for all your help.

think something like this will do?

https://www.amazon.com/dp/B004S494WO?psc=1

I see it has the wind jammer which I'm assuming would be good for outdoor filming when there's wind.

This X-T30 is getting good reviews

Fujifilm X-T30 4K Wi-Fi Mirrorless Digital Camera (Body Only) - Black with 64GB Deluxe Bundle and Travel Photo Cleaning Kit https://www.amazon.com/dp/B07P4DDTTG/ref=cm_sw_r_cp_api_i_y0nFDbKFCVG15

At that price point, you may want to consider spending the money on a rig for your phone. Otherwise you could check out the new base model GoPro Hero.

I would not recommend a webcam necessarily or any traditional style handicam near that price.