Could be the incorrect needle for the machine. Also, vinyl is difficult to sew because its "stickiness" characteristic tends to grab and hold the thread during the stitching process and is more likely to cause skipped stitches. Look for a needle with a large groove to shield the thread from the fabric--perhaps a topstitch needle or even a leather needle.

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All About Needles

Tips and Techniques for Sewing with Oil Cloth Vinyl Laminated Cotton

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My first question would be, Did you replace the needle with one the same length as the one you broke? Second: Is it a new needle? If you answered no to either of these questions this could be the cause. Your needle should be size 135x17.Size 23 for a thicker needle, size 20 for a thinner needle. If replacing the current needle doesn't cure the problem. Then it's a timing issue. Try dropping the needle down a little bit and retighten it. This is a simple timing fix that will do one of two things. 1) Make the problem go away. 2) Make the problem worse.If it makes the problem worse then adjust your timing.