Having been working with and advising companies on how to get great customer feedback for many years, I have a couple of ideas you could try.
One of the key assets you have is your customer list. You already have permission to approach them, I assume, so reach out to some to start a dialogue.
If you are able to segment them by sales, you can approach your great, repeat, customers with a simple question such as "What is it that you and your dog like most about our brand?" and "Are there items you are looking for that you haven't seen in my shop? If so, what?" or "What were you looking for the first time you found us?" Don't blast them to everyone, don't "survey" them, just ask the occasional probing question to try and understand what the "job" was they were looking to do when they engaged with you.
Another possible idea is a have a monthly contest where you ask a question in your newsletter and enter anyone answering into a draw for a small prize. Then publicize some of the best answers and give a shout-out to the winner in the next newsletter.
At the end of the day, you want to simply start a conversation with your customers, so you can listen and react to what they have to say.
There's only so much to cover in this forum, so please let me know if you have any questions.
Steve

Hi Andrew, your clothing line offers a great way for four-legged and two-legged buddies to share a bond. I believe your social enterprise mission to support Kiva is wonderful and creative. So let's get creative in achieving great customer feedback.
Your company is newly launched so I advise that you garner product feedback while promoting your brand and mission. Note, I said product feedback in lieu of customer feedback. I believe growth is paramount to continuing to exist and increase income and fund your philanthropy.
Grab friends with pets to walk in a group at popular dog parks. Have a friend serve as a roving reporter to interview people for key input.
Hold a pop up in collaboration with a popular food truck to also expand brand awareness and gain feedback.
Yes, encourage customer input continuously because they can provide active user experience intelligence.
Just a few ideas to get you underway. Continued success and for further insights, let's chat.

Hello!

I have worked with numerous early-stage businesses to assist them with planning, developing their brand, gathering of market research and creation & implementation of marketing plans.

I have taken a look at your site. It looks like you're off to a good start! It would be helpful to know more about the feedback you're looking to receive. For example, if you are interested in gathering insights on client's apparel likes and dislikes, spending, etc., I would recommend connecting with existing clients. Do you currently provide feedback surveys? If so, this is a great way to gather insights and feedback. Also, I see you are active on social media. You can utilize it to conduct polls to A/B test types of clothing, color options and more.

I would be happy to share more thoughts about your site/brand as well as recommendations. Feel free to reach out to schedule a call with me. I look forward to hearing from you soon!

-Leandra