General synonyms:
* Avidly
* Anxiously
* Ardently
* Desperately
* Enthusiastically
* Fervently
* Intently
* Keenly
* Passionately
* With anticipation
* With gusto
* With relish
More specific synonyms:
* Impatiently (if eagerly waiting)
* Hungrily (if eagerly wanting something)
* Thirstingly (if eagerly desiring something)
* Voraciously (if eagerly consuming something)
The best synonym will depend on the specific context of the sentence.