Can I save image created in PowerPoint as a printable vector?

I disagree PowerPoint is not used for drawing. I created these characters using PowerPoint, using the Line tool and other shapes, adding the background from Pixabay.

If you design a vector image in PowerPoint using tools like text and shapes, as long as you do not apply any shadows, blur, gradient, beveling, etc, you can export as a PDF and it will export as a vector.

Any effects you apply are exported as raster. Without effects, your PDF can be opened in Inkscape (free) or Adobe Illustrator and edited like any other vector graphic.

Also please note that your PNG is a raster graphic, and if you used it in your design, this part of your logo will not export as a vector.

While EMF and PDF can both be vector formats, you say that your logo contains a PNG image. PNG is a raster image format, so when you export to PDF or EMF, you'll have a vector image with an embedded PNG. This is probably the issue your printer is having.

You cannot simply convert a raster graphic like a PNG file to a vector graphic by saving it as a vector format. You will need to either 1.) re-create the PNG aspects as a vector or 2.) use an application like Adobe Illustrator to trace it.

If you are looking for a free program for vector graphics, I'd recommend Inkscape. Inkscape does have limited tracing abilities, although your mileage will vary depending on your image.

[I feel that I have a moral duty to tell you that PowerPoint is not for drawing things. PowerPoint is for making dreadfully boring slideshows with cheesy star-swipe transitions and clip art circa 1998. Use the appropriate tool for a job. Use Inkscape, Illustrator, or even LibreOffice Draw to make your logo; use PowerPoint for the Annual Q4 Sales Working Group Planning Meeting presentation.]