I care about animal welfare and climate change but do not want to go vegan. Suppose I eat a typical American diet. How much money would I need to donate each year to the most effective charities to reduce animal suffering and climate change by an amount equivalent to going vegan for one year? What about going vegetarian?
It would certainly be an amount greater than you spend on animal products, and even then, that money donated to whatever charity you choose doesn't cancel out the effect of your vote for the things those charities are opposed to.
You have to realise that money isn't buying new ice caps or replacing animals, it's trying to affect change in society and behaviour of people.
Sooner or later the charities you've supported are going to come to you and say 'go vegan', because reducing demand for these things is the only way to make them go away, and you can't put a price on that.
The animal welfare portion of this question already has a complex answer, before even beginning to consider the complexities of climate change and carbon offsets. Therefore in my answer I will focus on animal welfare and ignore the climate change component of the question.
This question implicitly assumes that vegans make up a relatively small portion of the human population. As a thought experiment, let's flip that on its head and assume that everyone in the world (except you) has already decided to go vegan.
In this scenario, what will be the effect of your donations to vegan advocacy groups? These organizations have one task left as their raison d'être: to convince you to adopt a vegan lifestyle. They run advertisements and people knock on your door, but you remain unconvinced. A collective realization dawns: the only solution is to find your price and pay it. Now it's your turn. Name your price, the amount of money that would convince you to adopt a vegan lifestyle forever, and you have the beginning of an answer to your question.
Let's return to the real world where vegans are indeed a minority. Which is the more effective way to reduce suffering -- reducing demand for animal products, or changing the way animals are raised and treated on farms? It is not easy to find an equivalency here because suffering itself is difficult to rank or quantify. And is death considered morally neutral in this question? (In Animal Liberation, Peter Singer argues that suffering is a well-defined concern, but dying is not.) This kind of question divides animal welfare organizations (eg. RSPCA, Humane Society of US) and animal liberation/animal rights organizations (Animal Defense League, FARM USA) to this day. No amount of math can definitively answer this for you.
So I'll assume the only scenario in which we can easily set up an equivalency -- you want to pay money to an advocacy group so that they can convince somebody to go vegan on your behalf who otherwise would not have done so (demand-side conservation). How much would that cost? Now we are getting close to a well-defined question! But even for organizations that work in this space, that number is quite elusive. Most people do not change their lifestyle after a single interaction with a brochure or a public demonstrator, so the real cost of "making" a vegan is amortized over an unknown number of interactions. And for any given campaign, the number of people who adopt a vegan lifestyle is also likely unknown, unless every person is identified and has a personalized follow-up. Animal Charity Evaluators goes into much more detail about the challenges in estimating these costs. I tried (but was unable) to find numbers from Animal Charity Evaluators, Animal Equality, Faunalytics, and the Centre for Vegan Advocacy.
But it turns out this approach is probably not optimal anyways. Recent findings have suggested that it is more cost effective to convince two people to go 50% vegan than to convince one person to go 100% vegan.
Even if this number were readily available, the problem is further compounded by the fact that most people who originally adopt a vegan lifestyle do not maintain that change, and revert to a diet that more closely resembles the standard diet for their culture and geography. To compute how much it would cost to "maintain" one additional vegan would need to take this into account as well. These are the type of questions that "effective altruists" (a group/movement) are constantly trying to answer.
Effective Altruism is based on utilitarianism, which has its own moral quandaries. The trolley problem is one famous example. If there was a scenario where five people could be saved by murdering one, would it be desirable or morally repugnant? This is not purely theoretical -- look closely enough, and you can see many examples of this moral dilemma present in the world today.
In this answer, I hope to have convinced you that this question is too broad and too complex for the answer to be distilled down to a simple number.
You need to decide for yourself if you value the welfare and lives of animals more than you value your unchanged lifestyle. Then live with that, rather than asking how much it costs to pay away your sins.