Star Valor has partial controller support, so you will have to use the mouse for menu interactions, and the keyboard for some things, but it's possible to limit that, and this guide suggests how. I'm disabled and can only play games like this with a controller; my right hand is screwed, too, so my mouse is left-handed which makes rebinding keys essential.

Conventions

I'll try to use the Xbox controller button names, although I play with an 8bitdo Switch controller, because Windows assumes all controllers are Xbox controllers and there's ghastly shenanigans with not-Xbox controllers that make life hard for Laious. Besides, the game tells you to use Xbox named buttons.

LStick/ RStick: left stick / right stick
RStick button: right stick button (press the stick until it clicks)
LTrigger/ RTrigger: left trigger / right trigger
LBumper / RBumper: left bumper / right bumper
select: the button to the left in-between the sticks
start: the button to the right in-between the sticks

Note
The game observes the Windows accessibility mouse pointer settings, although it does use custom pointers. For example, my mouse pointers are very large so I can see them; for me, the game also has huge mouse pointers which makes them super-easy to find. This is great. Thank you, Laious!

RTrigger button: Target enemy.
RBumper: Fire the starter laser.
LBumper: Fire the starter mining laser.
To target an enemy: RTrigger button
To fire the starter laser: RBumper
To fire the starter mining laser: LBumper

By default, when you add a weapon to your ship, and it's a good idea to do that ASAP because the starter weapons are miserable, it's bound to Mouse0. You have to change this and (for now) can't change the default (making it default to the controller is on Laious' long list). Select the new weapon and use the 'Change key' button to the right of the list of equipped weapons. Just press the button you want to use to shoot that weapon: the RBumper and LBumper work fine for me.

Rstick: turn the ship to face the direction of the right stick, indicated by the pointer
LStick: apply thrust in the direction of the stick
D-pad down: come to a stop

Turning is better in a small ship than a big one. Use gyroscopes, large gyroscopes, and crew skills to improve your turning speed. Thrust is increased by the thruster and by speed boosters. Mk II and III boosters are for larger classes of ship. If the ship is not facing the same direction as you're applying thrust, then (large) lateral thrusters help.

D-pad Left: increase thrusters
LTrigger + D-pad Left: decrease thrusters
D-pad Up: increase shields
LTrigger + D-pad Up: decrease shields
D-pad Right: increase weapons
LTrigger + D-pad Right: decrease weapons

'Increase' goes up from 0 to 5 pips, gives you more thrust/shield/damage, but it's less energy and heat efficient. The percentage increase in energy cost is shown in the right side, with the pips, and by default it goes up to 200%. The maximum benefit, meanwhile, is only 100%.

'Decrease' conversely reduces the number of pips, gives you less thrust/shield/damage, and it's more energy efficient.

X: dock at a station / inventory elsewhere
Y: open/close the sector map
A: clear target
B: scavenge

The remappings I've made that make things a little easier for me; YMMV!

Target enemy: RTrigger
Target friend/neutral: RStick button
Galaxy map: start
Cloak: select
Attack this target! A button (usually 'F' on the keyboard.)

Tell me in the comments. I'm all ears! Seriously, I'm very bad at seeing errors now, and you can never edit your own work. At time of writing I'm away from my computer for a couple of days, so I can't remember some defaults and I'll fill them in later.

This game has a solo dev. Expecting extensive accessibility features on release is a recipe for never getting a game released, and we should beware letting the perfect be the enemy of the good, but there are some areas controller support could be greatly enhanced, and an accessibility content patch would be fabulous.

• LTrigger modifies the D-pad, but AFAIK doesn't modify the ABXY buttons, and I'd like to see that possible. Y opens the sector map, having LTrigger+Y open the galaxy map would be ideal.
• Similarly there are keyboard bindings for your skills, knowledge and faction rep, and having them mapped to LTrigger+XAB.
• The galaxy map should pause the game if a controller is used. Being able to warp to safety is a crucial and common practice, but for accessibility, not pausing is terrible. For most able-bodied
  people, it takes less than a second to warp to safety when a system is already chosen, and little more if one isn't. Controller users have to switch to the mouse and click which can be slow and difficult. For me, my reactions are slower than normal anyway, I'm visually impaired and have a tremor, so it often takes several seconds to warp, as many as 5, which is plenty of time to get you killed.
• Being able to use the controller in the dock/inventory screen. I would imagine:
  LBumper/RBumper navigates the tabs of the station (and I would distinguish the active tab by making the inactive ones have a slightly darker background and light grey text).
  
  LTrigger/RTrigger navigates the tabs of the active column, either what's in my ship or what's in my hangar or inventory. I'm not sure if there's a use case for the tabs ever to be different, so maybe just sync them.
  
  Move up and down a column or between columns in the hangar with the D-pad or LStick. Ignore the buttons.
  
  A: equip/unequip (+LTrigger for all)
  B: deposit (+LTrigger for all)
  X: sell (+LTrigger for all)
  Y: destroy (+LTrigger for all)