I’m confused by this for a couple of reasons. I think you are using a Link to Screen button to get to the Members sheet? A link to screen button does not provide any way to connect the record you are on to the sheet the button is taking you to. Also your members sheet shows a column value for both Race 1 and Race 2.
I think you need to create a new sheet that contains an email, a race, and a Color column. The you need to create a relation column from the race to the new sheet. Then create a multiple relation column and and inline list component that uses the relation and is filtered by signed in user. This will take them to their own race record to mark the color they want for that race.