Part numbers
First, know that there’s different numbers that identify a part.
The LEGO Group uses two numbers: a Design ID (4 or 5 digits) that codes for the shape of the part and an Element ID (7 digits) that codes for a combination of shape and color. The Element ID is the number printed in the list of parts at the end of official instructions.
BrickLink and Studio use an Item Number: it’s generally based on The LEGO Group’s Desgin ID but as the company may use different numbers for the same part or the same number for different variants and as BrickLink prefers to keep the same base number for plain and decorated parts, they are often different.
For now, Studio only uses the BrickLink Item Number. So search the part on BrickLink to be sure of its ID, or try searching it by its BrickLink name:
Alternate Numbers
Some parts have more than one ID number, depending on the source. The LEGO catalog, BrickLink, and LDraw each use their own numbering systems, and the same physical part may have different IDs across all three.
If searching by part ID isn't returning results, try the alternate ID from a different source. For example, if a BrickLink ID isn't working in Studio, look up the LEGO catalog number for the same part and try that instead.
This is a normal part of working across platforms. Building familiarity with where to find alternate IDs and which ones Studio responds to will make your workflow much smoother over time.