No - Metadata Search app is natively built on the Salesforce platform itself and does not use any of the external logging utilities like Log4j, hence it's not affected by the Log4j vulnerability.
PS: View Metadata Search, Salesforce quick search app on Salesforce AppExchange: https://appexchange.salesforce.com/appxListingDetail?listingId=a0N30000009wgF7EAI