Yes you can. Litters or individual puppies are frequently abandoned at or handed into shelters and rescues. Often these puppies have been bred without any forethought and end up unwanted; the litter is larger than was expected, no buyers were found or the breeder just cannot cope. Some of these puppies are already 'second hand' having been bought from unscrupulous puppy farmers as a specific breed or size but in a matter of weeks have grown into a variety of dog unwanted or unsuitable to the new owner.
Fortunately, dedicated staff and volunteers in rescue and rehoming organisations pick up the pieces and provide the puppies with socialization, training and care until loving new homes can be found. Reputable rescues will vaccinate, microchip, treat for parasites and where appropriate neuter the puppy prior to adoption.
For a rescue puppy, call or visit a rescue and rehoming centre or organization. Use Rescue Review directory which you can find here.
To find the rescue centres or rehoming groups in your post code, county or region. Read the reviews here to gather useful information on which rescues offer the best services, help and support.