Here is a gallery of customizable items in Animal Crossing New Horizons. All of these items can be customized with a custom design pattern. The list below is ordered alphabetically.
You can also view custom designs made especially for these items by browsing their specific tags or category:
Beach Towel 
Blue Corner 
Bunk Bed 
Changing Room 
Clothesline 
Clothesline Pole 
Cushion 
Digital Scale 
Director’s Chair 
DJ’s Turntable 
Doghouse 
Drinks Machine 
Drum Set 
Effects Rack 
Electric Guitar 
Floor Lamp 
Floor Seat 
Frozen Bed 
Futon 
Giant Teddy Bear 
Hammock 
Iron Hanger Stand 
Ironing Board 
Ironwood Bed 
Kimono Stand 
Kotatsu 
Loft Bed with Desk 
Log Bed 
Log Chair 
Log Decorative Shelves 
Log Dining Table 
Long Extra-Long Sofa 
Log Garden Lounge 
Log Round Table 
Loom 
Low Screen 
Magazine Rack 
Mama Bear 
Neutral Corner 
Outdoor Bench 
Outdoor Table 
Papa Bear 
Paper Lantern 
Pet Bed 
Pile of Zen Cushions 
Plain Wooden Shop Sign 
Poolside Bed 
Record Box 
Red Corner 
Refrigerator 
Retro Gas Pump 
Rock Guitar 
Scattered Papers 
Simple Panel 
Stall 
Standard Umbrella Stand 
Table with Cloth 
Upright Locker 
Vintage TV tray 
Wedding Welcome Board 
Wheelchair 
Wooden Double Bed 
Wooden Mini Table 
Wooden Simple Bed 
Wooden Table 
Writing Chair 
Zen Cushion
Partial source of information is from the Animal Crossing Wiki. Images are from Nookazon.