The failure occurs when trying to connect to the VAP Wi-Fi network. I'm not even getting to the point where I can use VNC.

When trying to connect my phone or tablet to the astroberry VAP, I enter the default password of "astroberry" and the Android devices will authenticate the password, then cycle between these two states forever (see JPGs).

Is the WIFI password 'astroberry' or 'astroberry)' ?

It's just "astroberry". I know I have the right password because I get an authentication error if I enter anything else.

Have you tried using other devices (PC, laptop)?

It sounds like a security protocol issue - WPA etc - being out of date with latest wpa/wpa2 spec. Although I do not have a Samsung Tab this sort of behaviour occurs on other kit - i,e, can see network ,password correct but fails to connect. Alternatively it could be your browser!

So

1. Try www.samsung.com/au/support/mobile-device...ect-to-wifi-network/

2. If (1) fails to solve the problem try switching off security WPA etc altogether and see if that connects ok.

3. if (3) fails try connecting RPI4 to your local WiFi network as client then Web Browser from Samsung Tab S2

Good Luck

OK, I was able to connect to the VAP using my PC and my iPad, but it still fails with both of my Android devices (Tab S2 and S9). I suppose I could use my iPad in the field, but I'd really prefer to use my tablet as it has a larger and higher resolution display.

Last Friday, I reset my cell phone to factory defaults and went through the painful process of reloading and reconfiguring all the apps. That didn't help. I've used my cell phone and tablet in literally dozens of different locations all around the world and never had a problem connecting either of them to Wi-Fi, so this really has me perplexed.

Raspberry Pi 4 is very new hardware, the same for raspbian operating system. You can probably play with Hot Spot configuration so it matches your devices' needs.

I'm not sure what you mean by that. I'm fairly sure this isn't a hardware issue since my PC (Windows) and my iPad (iOS) can both connect to the astroberry VAP. But neither of my Android devices can even though they have never had a problem connecting to any other Wi-Fi network (and there have been dozens of them). That smacks of a software or configuration issue with the astroberry implementation. I just don't know where to even begin looking.

The way WiFi and BT work on Raspberry Pi is dependent on the chipset that Raspberry Foundation has chosen and firmware they provide with Raspbian system. There can be bugs in either of these components, including drop of WiFi transfer rates if you use 4K resolution over HDMI.
What you can do is to play with configuration of the Astroberry Hot Spot so maybe you can find the way your android devices connect with the hot spot (BTW, I use google pixel with vanilla android system and have no issues with connecting to it).
Righ-click WiFi icon on your top taskbar and select Edit Connections, then double-click Astroberry Hot Spot - you can play with settings on Wi-Fi (band, channel) and/or Wi-Fi Security tabs. Note that you should make changes while connected to screen and keyboard. Otherwise you can loose you connection.

Thanks for the advice. Interestingly, after disabling security it connects immediately. But if I enable any type of security (WEP or WPA), it fails to connect. I suppose I could run with it open but I'm not very comfortable with that since I plan to start using an astronomy site that will have a lot of other people nearby who are also going to be using telescopes and astrophotography rigs - it wouldn't be very good if someone connected (either accidentally or on purpose) and messed with my RPI.

So the problem is definitely related to Wi-Fi security and Android, I just don't know how to sort that out.

I run into the same problem. I cannot join the WiFi from my Samsung Note 8. My windows PC and an old Asus Android tablet can connect, but that Asus tablet have such an old Android version that I cannot install Kstars Lite on it.