I've selected hardwoods from a local family owned company (in business 42 years). Everything I can find out about them (BBB, Yelp, Facebook, etc) looks good.
But, just found out that they require 100% payment up front. Is this normal? Common?
I asked about it and they explained that they had issues with people paying the 50% up front and then the customer would go dark on them and not respond to calls and such.
Red flag?

They're saying that you should trust them but they don't trust you. If you want them to do the work then ask if they would accept payment that would mirror the % of work that has been done. Obviously you don't feel entirely comfortable with their way of doing business. Hardwood flooring installation isn't exactly brain surgery look around for other installers that do quality work.
I would wonder how many have stiffed them and why....have any dissatisfied customers done any on-line complaining. I would also wonder if any customers have actually not paid the balance due.....could just be a 'line' to get all the $ upfront. 'Local family owned' sounds good and I'm sure that you would like to support a local 'mom and pop' business....from personal experience I have done business with 'local family owned' businesses and I have regrets.....one business has been family owned for 130 years.

Typically we do 50% upfront when the order is placed, so that inventory can be reserved for the installation and not sold to other customers who may be able to pay 100% and take the material the same day. 90% required before delivery to site and installation begins, with 10% held back for after completion. There are definitely flaky clients out there who have tried to weasel out of paying that last 10% by making false claims about installation method, saying the material is defective, etc. It can sometimes take months to get that last 10% out of people. So I sympathize with 100% upfront - especially if the material lot is custom ordered and wouldn't be useful to them to resell should you decide to cancel and not pay at the last minute. I have clients that have orders from years ago where they paid 50%, we ordered the material, and they disappeared never to be heard from again, and I'm left with extra material in my warehouse that is taking up valuable space, and can't be easily sold (IE 400sqft of custom finish artisan engineered wide plank).

I'd ask if they feel comfortable with a 10% hold back to ensure quality of the installation. I imagine most places would.

Gotta pay a box store upfront so........

It's reasonable to pay for materials up front. But for labor I'd pay when the job is done.

For actual flooring material , I require 100% payment upon delivery . For labor I break it down to two payments of 50% . For a very large job that takes longer , I might break labor up into three equal payments .