Creating an index on table A but with reference to table B
If I understand you correctly you have two tables (for this question):

CREATE TABLE Clients
(Client int, ClientName varchar(50), Lots Of Other Columns)


and

CREATE TABLE Payments
(PaymentID, Payment Columns, ClientID)


Obviously not actual code there to create your tables...  

And your application is somehow joining those tables behind the scenes. Probably on the ClientID. So your user runs a filter on "Acme Company" and behind the scenes SQL Server is doing a JOIN statement.

Again - pseudocode

SELECT TheDataIWantFromPayments, TheCustomerDataIWant
FROM Payments 
INNER JOIN Customers
ON Payments.ClientID = Customers.CliendID
WHERE Customer.ClientName = 'Acme Company'


So here is where having more information can really help. On first blush this could be a problem with a simple index create as a solution. Perhaps there is no index on the clientID in each table. Perhaps there is no index on the ClientName.

But there could be other factors at play, too. It is possible that the vendor issues this query as a contains search using the syntax of CompanyName LIKE '%Acme Company%' and then no index on CompanyName would help. It could be that the database design requires a lot of key lookups and an index isn't the solution.

Either way - 6 minutes for 15 rows is not acceptable. I would look at those tables and at least verify that the CompanyName and ClientID (or whatever the columns are that contain those values) are indexed on each side. 

I would also strike up a conversation with the vendor. Even just you adding indexes could be an action that violates your support contract. Perhaps there are considerations that they already have. And you shouldn't have to be working through someone else's bad design or poorly performing application.

You can gather some data to help you. Play with the PAL tool and look at your performance data on your system. If you have a development/test environment and you can recreate this, perhaps through playing with Extended Events or SQL Trace you can see the query that issued and then run the query showing a query plan to get a sense for what is happening. But I would definitely loop the vendor in to a question like this.