You get access to Innate Spellcasting when using shapechange
Spellcasting and Innate Spellcasting are both separate examples of special traits, as can be seen in the basic rules for monsters. There are even monsters that have both traits, for example, the Drow Mage (note how they even use a different spellcasting ability):
Innate Spellcasting. The drow’s innate spellcasting ability is Charisma (spell save DC 12).
Spellcasting. The drow is a 10th-level spellcaster. Its spellcasting ability is Intelligence (spell save DC 14, +6 to hit with spell attacks).
The shapechange spell explicitly states that you do not obtain "the Spellcasting trait" (capitalized even) and not "any of its available spells" or anything similar1 suggesting that only the trait that is called "Spellcasting" in particular is not obtained.
Further, Jeremy Crawford has also unofficially confirmed that these are supposed to be read as being different traits in this tweet exchange that is also about shapechange:
Staffan Johansson @DackeStaffan
@JeremyECrawford When the shapechange spell says you don't get Spellcasting, does that include Innate Spellcasting?
Jeremy Crawford @JeremyECrawford
Spellcasting and Innate Spellcasting are two different traits. #propernouns #DnD
Therefore, we can conclude that using shapechange grants access to the Innate Spellcasting trait along with any other special traits the creature has (except the Spellcasting trait).
1. Noted by Medix2