First of all, you're going to need an emulator, probably for Android, if you want to rip from your computer. Nox is my emulator of choice for ripping from FEH, for the following criteria: it can download the app from the Play Store, can actually play and interact with the app (which you'll need to do if you don't want to link to your FEH account), can enable and disable root mode, comes with a file manager, and has a shared folder where you can move files between your computer and the emulated phone. I'm using version 6.2.8.1.
Secondly, you'll need an image editor capable of converting or opening webp images. (This is very important for a reason you'll see later.) I use GIMP 2.10.8 for my submissions.
Third, though this is only required for map characters: this SSBP viewer will allow you to view animations for a map character and make icons for them. Extract the zip anywhere, but I personally recommend your downloads folder or desktop.
Open your emulator, go through setup, and install FEH. Make sure you do not have root enabled before you open FEH; As far as I know, FEH automatically blocks rooted devices from playing, though I don't know if this applies to the tutorial. Open FEH and link your account or play through the tutorial until you get to where you download additional data. There's a few gigabytes, so grab a snack or something and wait.

When all your data is done downloading, close the app and go into your emulator's settings and enable root mode. You'll need to restart the emulator. Now open your file manager and navigate to: /data/data/com.nintendo.zaba/files/assets/Common. (Your file manager may ask you to activate root access mode; allow it.)
This is where all that downloaded data went to. The Face folder contains portraits and the Unit folder contains the map sprites and important SSBP files, both sorted by game of origin. The Wep folder contains weapon sprites sorted by type, and the Unit_Accessory folder contains accessories.
Units shown when downloading additional content are packed in the .apk under assets/Common.
(Nox specific, there may be different but equivalent steps for your device or emulator.) Select the folders you want to grab and navigate to your shared folder at: /mnt/shell/emulated/0/Others. Then click the three dots at the bottom left and click Copy selection here. Now you can access these files in your computer's file manager!

Unit and Face folders are grouped by game of origin, then sorted by character according to the character's original release date in the game. For instance, Lyn sprites will appear before Fiora sprites, because the first version of Lyn was released before the first version of Fiora.
Normal pool versions are denoted by Normal at the end of the folder name, resplendent alts are denoted by NormalEX01, legendaries are denoted by Legend01, mythics are denoted by God01, etc.

Now that that stuff is cleared up, go ahead and look at one of the sprites you grabbed. Do you notice anything about the thumbnail? It's broken, isn't it? Not to worry, it's actually a WEBP image renamed to PNG... for some reason. Luckily, GIMP realizes this when you open it and it'll display correctly. (If you want, you can overwrite the image without actually doing anything to it, and it'll become a real PNG.)

Most sheets will only have 7 images in it: Face (neutral art), Face_FC (barracks icon), BtlFace (attacking), BtlFace_BU (attack preview), BtlFace_C (special activation), BtlFace_D (damaged), and BtlFace_BU_D (damaged preview). However, many Heroes characters (folders that start with ch00) will have additional variations of their Face texture.
TSR icons are made from the BtlFace_BU image resized to 125 pixels tall, then cropped to 148 pixels wide; except if the unit is a Duo Hero or Tokyo Mirage Sessions character, where the Face image will be cropped and resized to show both characters.

Most units will follow this format. Place Body_AllTex, Head_C, and other sprites in the same sheet.

Transforming units, like dragons and beasts, have a separate folder for their transformed state. Include these sprites in the same sheet.

Split Duo Heroes into their respective characters. For instance, Winter Marth (folder ch01_00_Marth_MF_PairMain) is separate from Winter Elice (folder ch01_00_Marth_MF_PairSub).

To make icons, you need the unit's SSBP file as well as the PNG files. Open the character's folder and the folder you extracted your SSBP Viewer zip to, then drag-and-drop the SSBP file onto ssbp_viewer.exe. It will open a preview window and an instruction dialog. You don't even need to convert the sprites to actual PNGs for this step; they can remain in WEBP format.
Sprites from "Idle" or "Ok" animations are generally preferred. Weapons are optional in icons.
Some newer units have sprite previews on the official Meet some of the Heroes page. These are also acceptable if the unit has one. Resplendent Heroes also have previews here.

The titles of map characters should match their portrait, except in the case of Duo Heroes.
"Seasonal" versions and additional normal pool versions of characters should be named with their introductory banner in parentheses. For example, "Camilla (A Sketchy Summer)", "Hector (Arrival of the Brave)", and "Eirika (Sacred Memories)".
Legendary versions have "Legendary" or their title in parentheses. For example, "Chrom (Legendary)" and "Ike (Vanguard Legend)".
Duo Hero portraits should be named in the following manner: "[MainChar] & [SubChar] (Banner name)". For example, "Ephraim & Lyon (Desert Mercenaries)" and 'Marth & Elice (Glorious Gifts)".
Duo Hero map characters, being split, should be named in the following manner: "[MainChar] (Banner name)" and "[SubChar] (Banner name)".