Existent is an adjective.
Existing is a past participle used as an adjective.
They are synonyms and interchangeable.
The word existent is more likely to be used in the combination form "non-existent customers", or "non-existent voters".
In the positive form you are more likely to come across "existing customers" or "existing voters", etc.
There is a slight difference betweent the two.
The "existing customers" may refer to the customers you have now, or to potential customers.
The "existent customers" is more likely to refer to the the actual customers you have now.