Yes, Aasimar can use this ability in an antimagic field
Only magic and magical effects are affected by an anti-magic field as indicated in the description of the spell:
Spells and other magical effects, except those created by an artifact or deity, are suppressed in the sphere and can't protrude into it.
Sage Advice gives us tests for determining if something is considered magical. I will go through and apply these tests to the Radiant Soul ability.
- Is it a spell? Or does it let you create the effects of a spell
that’s mentioned in its description?
It is not a spell and no spell is mentioned in its description. See this Q&A for a description of how to tell if something is a spell.
It isn't a spell or an attack so no.
- Is it fueled by the use of spell slots?
It uses no spell slots.
- Does its description say it’s magical?
Nowhere in the description does it say it is magical.
The Aasimar ability Radiant Soul passes none of these tests and is thus not magical. Since it is not magical it is not affected by antimagic field.