The Pet Animals Act 1951 dictates that keepers of pet shops must have a licence, and Lucy's Law bans third-party puppy & kitten sales, but many who sell pet animals are excluded from these requirements due to infrequent sales or selling of older dogs. This loophole makes enforcement much harder.
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If all sales required a license this would make it much easier to spot those not adhering to this. It would also make it easier for members of the public to check their seller before a purchase, rather than relying on intuition.
This would not negatively impact responsible pet sellers as these are already required to have a licence. The fees for licences could go some or all of the way towards covering costs for license issuing and continued monitoring, including regular microchip checks.