Good Morning Martha, 

Thank you for contacting the U of M Extension Service. 

I did some research on storing martagon lily bulbs over the winter. We do not endorse any particular business. 

Please click on the following link from Iowa State: https://www.extension.iastate.edu/linn/digging-and-storing-bulbs-over-winter#:~:text=A%20sharp%20shovel%20and%20a,for%20spring%20by%20digging%20now.

This information states: 

The general rule of thumb for storing bulbs is:

1. Do not injury the bulbs when digging. Dig far enough away from the stem so you will not cut into the bulb.
2. Only shake the dirt from the bulbs do not wash them. The exterior of the bulb needs to perfectly dry before storing.
3. The bulbs should not touch each other in storage. If one bulb rots the bulb touching it will also spoil.
4. Store the bulb in a 40-50 degree cool dry area. A garage, basement or root cellar will work.
5. Although the bulbs should be dry on the exterior the bulb should never be allowed to dry out. Some bulbs will need to be stored in peat or vermiculite to keep the necessary moisture in the bulb.

Here is another link from Brecks that indicates they should not be stored for more than three weeks  in a refrigerator: https://www.brecks.com/pages/how_to_tips_growing_instructions_lilies#:~:text=If%20you%20don't%20have,bulbs%20have%20developed%20a%20sprout.

I have also attached at document from the Lily Society about martagons. This document states: 

Martagon bulbs planted in the spring generally will not flower in the first year and may not even emerge above ground. They may stay dormant until the next season. If all goes well these individual bulbs will develop into major clumps of vigorous martagons over  the next 5 - 10 years. In some gardens and in some locations (including my own garden) some martagon types unfortunately will multiply and grow very slowly or hardly at all.

Good Luck!