Note: This is taken from reddit and not my guide, I did use it to successfully invite 8/10 villagers though so I thought it was worth more exposure.
Disclaimer: I wouldn't make this guide or suggest using this method if the cards were available at retail, but I think it is far given the current price gouging by online resellers.
A Samsung device running Android - there is currently no support for the Amiibo feature on non-Samsung devices, there may be in the future. There will likely never be iOS support due to Apple's stance regarding their devices.
The JoyCon Droid (Public Alpha) emulator app - This can be downloaded from the Play Store, beware the free version has video and audio ads. You can remove them by paying for the app, but otherwise it is fully functional for our purposes with the free version.
The Amiibo card data - I can't list where to get these, but a simple google search will point you in the right direction. These are commonly called "banks" and the term may help your search.
Download the Amiibo card bank. They often have every card, separated in each folder by series. I suggest taking a few minutes and choosing which villagers you want to bring into the game, and creating a new folder that is easy to access. The Amiibo function is AC:NH is timed, so if you take too long navigating to the correct folder or digging through 100 files for the one you want can lead to frustration.
Once you have your new folder setup with all your favorite villagers, you need transfer the folder onto your phone. You can just plug your phone directly into your computer via USB and allow access to the device files when the prompt comes up on your phone (just a few seconds after plugging it in). Drag and drop your folder into an easy to find directory on your phone (mines just in the root directory of my SD card). Now we're ready to pair our phone to the Switch!
Install the Joycon Droid app on your device via the Play Store. The first time you open it, there will be a quick guide on how to pair the app to your Switch. Before you pair, make sure to go into the apps main page and click Settings, then toggle the Enable Amiibo switch to allow the app to use the Amiibo feature.
When pairing, make sure to use the "Joy-Con" Pro option, not the Left or Right cons. I tried only using the right con the first time and it disconnected my phone. Your phone will request access to Bluetooth, this is how it communicates. If your Switch asks to update the controller software, DO NOT DO THIS!! Finally, you may also get a pop-up on the app asking to enter a MAC address, you can just dismiss this, we don't need it. Once you follow the guide on the App and have paired your phone, your ready to start sending invites!
Note: you must use the emulator to access the Nook Amiibo terminal, so its best just to use it as you would a normal controller while your inviting villagers.
You can do all the pairing while in-game, so don't worry about opening or closing the game in-between disconnects or invite time-outs, just re-pair the emulator and try again. When you are prompted to place the Amiibo on the joycon, tap the center black button on the Emulator controller and it will open your file directory. Open your villager folder and select the villager file you wish to invite. Remember to try and be quick to avoid a timeout. That's it! You've done it
I hope this helps you avoid spending a ton of money on cards, and you are able to enjoy NH to the fullest