General synonyms:
* Foraging
* Rummaging
* Scrounging
* Searching
* Hunting (in the sense of searching for food)
* Picking
* Collecting
* Gathering
More specific synonyms:
* Raiding (when referring to scavenging for food or supplies)
* Salvaging (when referring to recovering usable materials from discarded items)
* Junk picking (when referring to scavenging for discarded items)
* Dumpster diving (when referring to scavenging through trash bins)
* Scavenging (this word is the most direct synonym and is often used in a literal sense)
The best synonym will depend on the specific situation you are describing.