You can use the GET method of the sObject Rows by External ID resource to get records with a specific external ID.
The following example assumes there is a Merchandise__c custom object with a MerchandiseExtID__c external ID field.
Example usage for retrieving a Merchandise__c record using an external ID
curl https://MyDomainName.my.salesforce.com/services/data/v68.0/sobjects/Merchandise__c/MerchandiseExtID__c/123 -H "Authorization: Bearer token"
Example request body
none required
Example response body
{
"attributes" : {
"type" : "Merchandise__c",
"url" : "/services/data/v68.0/sobjects/Merchandise__c/a00D0000008oWP8IAM"
},
"Id" : "a00D0000008oWP8IAM",
"OwnerId" : "005D0000001KyEIIA0",
"IsDeleted" : false,
"Name" : "Example Merchandise",
"CreatedDate" : "2012-07-12T17:49:01.000+0000",
"CreatedById" : "005D0000001KyEIIA0",
"LastModifiedDate" : "2012-07-12T17:49:01.000+0000",
"LastModifiedById" : "005D0000001KyEIIA0",
"SystemModstamp" : "2012-07-12T17:49:01.000+0000",
"Description__c" : "Merch with external ID",
"Price__c" : 10.0,
"Total_Inventory__c" : 100.0,
"Distributor__c" : null,
"MerchandiseExtID__c" : 123.0
}