There is a real discomfort in asking for a referral, and it has a concrete cause: almost every request, the way it is usually made, transfers work to the other person.
"If you hear of anyone who needs it, let me know" sounds polite and light. In practice, it forces whoever heard it to do three things: remember what you do, mentally search their own network and decide on their own who fits. That is a lot for a free favour, which is why it almost never happens.
What makes a request easy to answer
Be specific until it hurts
Compare "I am looking for new clients" with "I am looking for food manufacturers in inland São Paulo that package in glass". The first sentence triggers no memory at all. The second makes the person think of someone concrete in five seconds, or quickly conclude that they do not know anyone, which is also a useful answer.
Ask for a person, not a category
"Do you know anyone like that?" is easier to answer than "can you introduce me to the food industry". One question is answered with a name; the other requires the person to put together a plan.
Make the next step easy
Offer the text ready to go: "if it makes sense, I can send you two lines for you to forward". That takes away the task of explaining what you do, which is exactly the part where referrals usually go wrong.
The order that removes the awkwardness
The discomfort disappears when the request is not the first thing. Two practices solve almost all of it:
Give first. If you have already introduced two people in your network, asking stops being a favour and becomes an exchange already under way. It is not calculation, it is how relationships work.
Ask first. Before saying what you are looking for, ask what the other person is looking for. Besides being more pleasant, that usually reveals the real fit, which is often not the one you had imagined.
When not to insist
If the person did not reply to the request, it should not be repeated in different words the following week. Referrals depend on the reputation of whoever gives them, and nobody risks their own out of politeness. Silence usually means "I do not know anyone" or "I do not know you well enough yet", and both are resolved with time, not with insistence.
Where the request works best
In small, recurring groups, where people already know what each one does, the request stops being a favour and becomes routine. It is the logic of bringing a concrete need to the group every week, instead of waiting for a coincidence.
The business networking guide deals with how to build that relationship base, and the page on how a business round works shows the format in which those conversations happen with a set schedule.