Photoshop elements is a pixel editor with only Shapes and Type being vector, so you really can't do much in terms of vectors with photoshop elements. The only "vector" file format photoshop elements can open or save vectors in is pdf.
If a subscription to Adobe Illustrator doesn't interest you, there is Inkscape an open source vector application.
You can save layered files in Photoshop Elements (so that includes any shape layers and type layers that you might have created) as a PSD, then Import that PSD into Adobe XD. There you'll find that your shapes are still editable vectors and your type is still editable too, along with a whole bunch of standard vector tools if you want to further tweak your design. When you've finished, you can export it as an SVG.
There's even a plug-in (Vectorize) which does an excellent job at converting pixel-based images into vectors 🙂