There are a few ways to go about this.
One is very programmer heavy. There's a lot of data processing that goes into machine learning, but you can't quite separate the two. Often, the programmer who prepares and processes the data needs to write the code that actually runs the model and parses the result, and this means you benefit from more understanding of machine learning. That role is most likely the best one available to programmers who don't have much mathematics background in this area.
To really get into machine learning itself as a data scientist, though... I do think it requires some math. There's a reason a large percentage of people who work in this field have an MS or PhD in a very quantitative field. And I don't just mean algorithm designers - to really be able to explain the difference between naive bayes, random forest, neural nets, and logistic regression, it helps enormously to have a background in math.
To illustrate this, I've taken two coursera courses on data science. They were both excellent, but approached from different angles. Bill Howe's data science class involved an exercise to use a random forest to do some classification, but the focus was on calling the scikit-learn library. We did of course review the algorithm, but not in mathematical depth.
Andrew Ng's course on machine learning got into implementing the algorithms (with a language called Octave, which honestly I didn't like much, but that's a completely different topic where plenty of people would disagree with me). To do that class, honestly, I'd just ask if the terms "vector calculus", "matrix of second order partial derivatives" or "logistic function" mean something to you. It's ok if you can't define these things on the spot, but was there a time when you could? You can get up to speed, but I'd say if you haven't taken basic calculus through differential equations (with linear algebra), then you won't be able to understand this material.
I've been impressed with how well people learn on their own, picking up a lot of math as they go along. And I don't think you need to be able to implement these algorithms yourself to use them meaningfully. But if you're going to be deciding what kind of model to use, even if you're using libraries to do it (and most people who can implement these algorithms would still use a library), I think that you do need to be able to describe how a neural net work vs random forests vs logistic regression vs naive bayes. There is a side of this that is very math-y as well.
On the bright side, we live in an era of amazingly available learning material. I personally think a dedicated person can probably learn calc, linear algebra, and differential equations through web-based coursework now.
SO overall, I'd say - start on the data side as much as possible, leveraging your programming skills. While doing this, keep getting more exposure to ML algorithms, and make sure you are taking a coursera or other web-based class on the side.