Can an individual submit claims for Disability Income Insurance (DII) from 2 different insurers?

You talking about which product/policy for DII? if careshield life supplements then limit is 5K per insurer, that is if the various insurer allows you to buy more after you max out 5k with the first insurer.

Assuming that you have not hit the maximum monthly benefit (that aggregates across the various insurers), then yes, you are able to submit a claim with both insurers.

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Otherwise, the payout will likely be subjected to the financial limit rules that the insurer has; i.e. you will receive a maximum payout and not anything more than that.