From v7.3, you can follow the steps below to assign the necessary user permission to use the Metadata Search.
Step 1 - Assign the managed (existing) Metadata Search Permissions permission set. [Required]
- Go to Setup -> Permission Sets -> Click on “Metadata Search Permissions” -> Click on the Manage Assignments button.
- Assign this permission set to desired users.
Step 2 - Create (and assign) a new permission set labeled Metadata Search Permissions (with API Enabled). [Required]
Please create a “Metadata Search Permissions (with API Enabled)” permission set and assign this permission set to desired users.
- Go to Setup -> Permission Sets -> Click on Create button -> Enter label as “Metadata Search Permissions (with API Enabled)” -> Click Save.

- Click on “System Permissions”.

- Click Edit -> Mark “API Enabled” & “Lightning Experience User” checkboxes as True -> Click Save.


- For users needing search access to set up and config entities like Flows, workflow rules, etc., mark the “View Setup and Configuration” checkbox as True-> Click Save. The “View Roles and Role Hierarchy” permission will be automatically checked.

- For users needing search access to Buttons and Links, mark the "Customize Application" checkbox as True-> Click Save.

- Now assign this new permission set to desired users.
Step 3 - Enable Lightning Web Security [Required]
- Please enable Lightning Web Security.
- Go To Setup -> Session Settings -> Lightning Web Security
- Mark Use Lightning Web Security for Lightning web components checkbox as True.
Step 4 - Add remote site setting for classic UI [Required for classic UI]
- Go to the Salesforce classic UI and open the Metadata Search tab.
- Could you copy the URL as shown below?

- Go to Setup -> Remote Site Settings -> Click New Remote Site -> Add remote site name and remote site URL with copied URL in step 2 above.

Please Note:
- View Setup and Config permission are required to search the following metadata entities:
- ExternalString (Custom Labels)
- FieldSet
- Flow
- LightingWebComponent
- Layout
- CustomTab
- ValidationRule
- WorkflowAlert
- WorkflowFieldUpdate
- WorkflowOutboundMessage
- WorkflowRule
- WorkflowTask
- Customize Application permission is required to search Buttons and Links.
- API Enabled permission is required irrespective of the search (metadata) entity.
PS: View Metadata Search, Salesforce quick search app on Salesforce AppExchange: https://appexchange.salesforce.com/appxListingDetail?listingId=a0N30000009wgF7EAI