No will

He is entitled to a portion of your mom's "stuff". Look up "Surviving Spouse Allowance". You can speak to an attorney or three in Indiana for more information.

For reference: 2021 Indiana Code :: Title 29. Probate :: Article 1. Probate Code :: Chapter 4. Surviving Spouse and Family Allowances :: 29-1-4-1. Surviving Spouse and Family Allowances

 

I'm curious why you think you can't do this. Just because you're not local doesn't mean you can't be appointed as executor/administrator.

Contact a probate attorney in the county where your mother lived at the time of her death and seek advice. It is possible that a demand letter will prompt husband #5 to probate the estate. If not, you'll want to discuss with an attorney what would be involved if one of you or your sister seeks appointment.

 

Indiana Intestacy Laws – heirbase.com
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https://www.scotthalelaw.com/post/indiana-intestate-succession-laws
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What is property division in an Indiana divorce?

Also known as equitable distribution, property division is the process of dividing property rights and obligations between spouses during the process of a divorce. Property division may be agreed upon between the soupses through a property settlement, or it may be decided in court during the judicial process of divorce. The process of property division is affected by state laws such as community property laws, definitions of marital contributions, etc.

Indiana is an equitable distribution state, and assets acquired both during and prior to the marriage can be subject to division following divorce. Some factors considered by Indiana courts in a property division case include non-monetary contributions, economic misconduct and a list of other factors defined in Indiana law. This page summarizes the most important aspects of property division laws in Indiana.

Indiana Marital Property Division Laws
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When I have had to do this job (for family members' estates), I've never had to appear in court (either personally or by video), but I have no personal knowledge about courts in Indiana.

FYI, it seems that most states will allow an out-of-state family member to serve as executor/administrator as long as that person has a local attorney or other representative available for service of any papers.