Uber Technologies Inc.’s recent stumbles over a string of unwelcome events are giving its ride-hailing rival Lyft Inc. a chance to seize the moment.

Lyft, which has marketed itself to drivers and riders as a kind of anti-Uber, has been quietly pitching investors on a new funding round that it hopes will net at least $500 million and a valuation of between $6 billion and $7 billion, according to people familiar with the matter.

It...