Absolutely. You do not have to use fusible web. There are several methods of applique: needleturn, where you cut out the shape and hand-sew it to the fabric, turning the seam allowance under with the needle as you go. Or you could use the template-and-starch method, where you use heat-resistant template plastic (or double layers of freezer paper) cut in your shape of choice, then press the seam allowance over the edge of the template. Or you could use the raw-edge applique method, where typically you'd fuse the applique piece to the fabric, then stitch around the shape using a blanket stitch or satin stitch. I've had success using washable school glue instead of fusible web. This way I can sew the applique patch on, then carefully pull apart the glued pieces and cut away the backing fabric.
Here is a series of how-to videos and instructions for applique, this quilter makes beautiful quilts. Hope this helps!
http://erinrussek.typepad.com/one-pi...lique-lessons/