Regarding what proficiencies one gets when multi-classing with a Voidrunner's Codex class, it should be consistent with the Multiclassing Proficiencies table on page 378 of the Level Up Adventurer's Guide.
I would suggest something along the lines below, with the assumption that someone may mix-and-match with LUA5E, so light armor proficiency matters though all Voidrunner's Codex classes get it.
- Psion: Light armor, medium armor, Arcana skill
- Psyknight: Light armor, medium armor, shields, simple weapons, blades, one Psyknight skill
- Scientist: Light armor, Science skill, two tool proficiencies
- Scout: Light armor, simple weapons, martial weapons, one Scout skill, one Scout tool proficiency
- Trooper: Light armor, medium armor, shields, simple weapons, martial weapons
Comparing Voidrunner's Codex vs. the LUA5E classes: Trooper is similar to Fighter; Psyknight is similar to Ranger; Scientist is similar to Bard; Scout is similar to Rogue; and Psion is similar to Druid.
Replying to Selganor's suggestions in the post above:
1. Both Psyknights and Troopers get shield proficiency, so I don't see a reason to omit that for multi-classing to Psyknight. I agree with you beyond that.
2. Regarding weapon proficiencies, Scouts also get martial weapon proficiencies, so should get that when multi-classing. I agree with you beyond that.
3. Regarding skills, Psions get Arcana + 3, Psyknights get 3, Scientists get Science + 3, Scouts get 3, and Troopers get 2. Consistent with 5E D&D and LUA5E, classes with 3+ skills should get one skill proficiency when you multi-class to them. (For example, compare Psyknight to Ranger.) To make the list of proficiencies above more balanced, I'd add an additional Psion skill proficiency and Scientist skill proficiency when multi-classing to them.
4. Regarding attribute requirements for multi-classing, yes Dexterity 13 would be fine for Scouts and Troopers. That gives us this updated list:
- Psion: Intelligence 13, Wisdom 13, or Charisma 13, as selected for the psionics ability modifier.
- Psyknight: Wisdom 13
- Scientist: Intelligence 13
- Scout: Charisma 13 or Dexterity 13
- Trooper: Dexterity 13 or Strength 13
I have simplified Psyknight to not require Dexterity 13 for multi-classing per my comment two above, as that permits builds with Strength/Wisdom, Dexterity/Wisdom, or purely Wisdom depending on your synergies and focus - e.g., a Soulblade build focusing on Wisdom.