Hi Chaucee,
Couple of ideas:
Write a grant - $5,000 is a relatively low ask and as a brand new business, you actually have a few different pools of grant orgs including new non-profit/seed money funders, animal welfare funders, and local or regional specific grant organizations. Being able to have a business plan that shows the need for this type of care (should be pretty easy!) as well as the number of animals and people that will be impacted, and a plan for keeping your organization sustainable without more grant funding will be key there.
Find an angel "investor" - Just like with a new startup, you can sometimes find someone you know, or are acquainted with, or are related to, to put in some money to start the project. While a startup would be looking for an investor who is interested in making their money back and more, you're looking for a like-minded individual who has the treasure to support your time and talent for the work. If it's not someone you know directly, maybe it's someone connected to another person in the organization, or another non-profit or community service group in your community. It might also just be a successful local business owner or local philanthropist.
You may also want to look at this as an opportunity to grow your board in the future. New non-profits generally only require three officers, and it's probably you and two other people you know, but think about who in your community might be a good, and well resourced, person to approach to help guide the organization and potentially contribute as well. There are absolutely people in your community who would donate more than $5,000 if you have a good pitch, an impactful mission, and a sustainable plan.
Crowdsource your project - This is probably a little easier once you have your animals and your stories, but if you can talk really, really well about your mission, why you want to start this, what stories of needs you've seen, the senior dog that inspired you, etc. Anything that really inspires people to want to be a part of your project, you can try recruiting through your personal Facebook and social media and encourage anyone you know to spread the word.
You need 500 people to give $10 or 250 people to give $20, in the grand scheme of social media that's also not a huge ask. You can accelerate this process if you can get any media coverage of your project, which depending on your market might not be that hard to do either. Generally you need a hook for TV or even newspaper, which is why having an animal makes a difference. Maybe have a dog or cat waiting in the wings who's story you can bring up as to helping get them to you for hospice care.
------------------------------
Jeff Okazaki
Humane Society of Jefferson County
------------------------------