Charcoal, 53 bytes

≔Ｅ³∧⊖ι⊘⁺⊖ι₂⁵υ≔ＥυＥυ§υ⁺κμυ⊞υＥυ¹ＦυＦＥ³Ｅι⎇⁼κν±μμＦ¬№υκ⊞υκＩυ 

Try it online! Link is to verbose version of code. Explanation: Uses the formula from this Math.SE answer.

≔Ｅ³∧⊖ι⊘⁺⊖ι₂⁵υ 

Get the point [1/ϕ, 0, ϕ].

≔ＥυＥυ§υ⁺κμυ 

Get its rotations around the axis x=y=z.

⊞υＥυ¹ＦυＦＥ³Ｅι⎇⁼κν±μμ 

Loop through each point and get its reflections.

Ｆ¬№υκ⊞υκ 

Save and process any new reflections found.

Ｉυ 

Output the final dodecahedron.

Charcoal, 53 44 bytes

≔Ｅ²Ｅ³∨ι∧⊖λ⊘⁺⊖λ₂⁵υＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκ⊞υκＩυ 

Try it online! Link is to verbose version of code. Explanation: Uses the formula from this Math.SE answer.

≔Ｅ²Ｅ³∨ι∧⊖λ⊘⁺⊖λ₂⁵υ 

Get the points [1/ϕ, 0, ϕ] and [1, 1, 1].

ＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκ⊞υκＩυ 

For each point, rotate it around the axis x=y=z, and also reflect the rotated point in the xy plane, and save and process any new points thus found.

Ｉυ 

Output the final dodecahedron.