TAIPEI—When buyers need chips in a pinch, they turn to Erik Drown, a middleman who is able to source scarce parts.

Mr. Drown’s decades of experience are now up against one of the worst semiconductor shortages ever. Asked recently by a client to help find 1,200 chips in a matter of days, Mr. Drown scoured his industry contacts. He couldn’t find all the needed chips but was able to get preliminary price quotes on about half.

It is a painful truth for businesses world-wide: They need chips but can’t find them. Making matters worse is that the typical problem solvers—chip brokers and middlemen such as Mr. Drown—find themselves in some cases just as perplexed and empty-handed. Yet they are busier and making more profits than ever.

“A lot of companies that would never go to someone like me, they have to because if they don’t, they’re not going to survive,” said Mr. Drown, global sourcing director at Select Technology Inc., an electronic-components distributor based in Rowley, Mass.

A breakdown in the typical distribution of chips adds extra layers of complication to a monthslong shortage that remains widespread and is likely to continue through the end of the year, say chip makers, wholesalers and buyers.