At will indicates you do not need spell slots.
The rules for cantrips help clarify this language of “at will”:
A cantrip is a spell that can be cast at will, without using a spell slot and without being prepared in advance.
To be clear, the spells from the Svirfneblin Magic feat are not cantrips, but the rules for cantrips give us a clear understanding of what “at will” means with respect to spellcasting.
This is also obvious from the fact that being a spell caster is not a requirement of taking the feat. If the feat required using slots for casting the spells, it would list that as a requirement in some way. The Eldritch Adept feat from UA 2020: Feats, while not official material yet, gives an example of how this would look:
Prerequisite: Spellcasting or Pact Magic feature
This feat would be totally useless to classes without spell slots, unless you can cast the spells without them.
Innate Spellcasting means it doesn’t use spell slots
In the Monster Manual, the Innate Spellcasting trait is given a definition for NPCs:
A monster with the innate ability to cast spells has the Innate Spellcasting special trait. Unless noted otherwise, an innate spell of 1st level or higher is always cast at its lowest possible level and can't be cast at a higher level.
In contrast, the regular Spellcasting trait is described as follows:
A monster with the Spellcasting class feature has a spellcaster level and spell slots, which it uses to cast its spells of 1st level and higher. The spellcaster level is also used for any cantrips included in the feature.
Now, you aren’t an NPC, but this demonstrates the common rules understanding of Innate Spellcasting: no spell slots necessary.