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You can, but the sorrel will usually outcompete the lettuce over time. I recommend keeping these separate, but it will not harm anything to try. Over time, though, the lettuce will not be able to grow once the sorrel takes over.

Can you send us an image? That would aid in identification.

Types of Sorrel
Garden sorrel (Rumex acetosa) is also called English sorrel or common sorrel. A perennial valued for its early spring greens, garden sorrel is available as a seed-sterile variety called ‘Profusion.’ Cutting off flower spikes to prevent unwanted reseeding is the biggest challenge to growing garden sorrel.

French sorrel (R. scutatus) has distinctly arrow-shaped leaves, and strains have been selected for low oxalic acid content. Like garden sorrel, French sorrel can become invasive if reseeding is not controlled.

Blood sorrel (R. sanguineus), also called red sorrel, makes a beautiful ornamental to grow in partial shade, but the leaves are only edible when very young. Some tangy mesclun mixtures include red sorrel.

https://www.gardeningknowhow.com/edible/herbs/sorrel/sorrel-herb.htm

Keeping spiders away, usually, consists of physically removing their webs, food source, and hiding spots. Keep in mind, removing spiders will mean that you will need to have a backup source of insect control ready to take over the job that the spiders are doing.

This article will help you to remove spiders: https://www.gardeningknowhow.com/plant-problems/pests/insects/spider-garden-pests.htm

This article will help you with organic insect pest control in place of the spiders: https://www.gardeningknowhow.com/special/organic/natural-home-pesticides-organic-garden-pest-control.htm