I agree with Mrs M. The dog's your way in. Unfortunate that we might all be in lockdown by the weekend, which might put the kibosh on your opportunities.
I would just make chit chat about the dog, and then say, 'Well if you fancy going for a walk with (said dog), let me know. There's a pub/cafe at xxxxx that allows pets'. Make it during the day; casual, no pressure.
That leaves it up to her to respond, but there's no pressure. The key thing is though, don't be too upset if she doesn't take you up on it. It might not be the right time for her or she might not be looking for romance. If you don't get a positive result, the worse thing would be to stop using the shop out of embarrassment or feeling rejected. You'll make her feel bad, when she shouldn't, and the incident will grow out of all proportion in your head. Just carry on as usual. Good luck.