You will need to join to your sub requester attribute table to the query twice. One with the attribute of Urgent and one with the attribute of Close.
You will need to LEFT join to these for the instances where they may be null and then reference each of the tables in your SELECT to show the relevent attribute.
I also wouldn't reccomend the cross join. You should perform your "OR" join on the personnel table in the FROM clause rather than doing a cross join and filtering in the WHERE clause.
EDIT: Sorry, my first response was a bit rushed. Have now had a chance to look further. Due to the sub requester and the sub requester attribute both being duplicates you need to split them both up into a subquery. Also, your modified date could be different for both values. So i've doubled that up. This is completely untested, and by no means the "optimum" solution. It's quite tricky to write the query without the actual database to check against. Hopefully it will explain what I meant though.
SELECT
r.RequesterID,
p.FirstName + ' ' + p.LastName AS RequesterName,
sra1.ModifiedDate as UrgentModifiedDate,
sra1.AttributeValue as Urgent,
sra2.ModifiedDate as ClosedModifiedDate,
sra2.AttributeValue as Closed
FROM
Personnel AS p
INNER JOIN
Requester AS r
ON
(
r.UserID = p.ContractorID
OR
r.UserID = p.EmployeeID
)
LEFT OUTER JOIN
(
SELECT
sr1.RequesterID,
sr1.ModifiedDate,
sa1.Attribute,
sa1.AttributeValue
FROM
SubRequester AS sr1
INNER JOIN
SubRequesterAttribute AS sa1
ON
sr1.SubRequesterID = sa1.SubRequesterID
AND
sa1.Attribute = 'Urgent'
) sra1
ON
sra1.RequesterID = r.RequesterID
LEFT OUTER JOIN
(
SELECT
sr2.RequesterID,
sr2.ModifiedDate,
sa2.Attribute,
sa2.AttributeValue
FROM
SubRequester AS sr2
INNER JOIN
SubRequesterAttribute AS sa2
ON
sr2.SubRequesterID = sa2.SubRequesterID
AND
sa2.Attribute = 'Closed'
) sra1
ON
sra2.RequesterID = r.RequesterID
SECOND EDIT: My last edit was that there were multiple SubRequesters as well as multiple Attribute, from your last comment you want to show all SubRequesters and the two relevent attributes? You can achieve this as follows.
SELECT
r.RequesterID,
p.FirstName + ' ' + p.LastName AS RequesterName,
sr.ModifiedDate,
sa1.AttributeValue as Urgent,
sa2.AttributeValue as Closed
FROM
Personnel AS p
INNER JOIN
Requester AS r
ON
(
r.UserID = p.ContractorID
OR
r.UserID = p.EmployeeID
)
INNER JOI N
SubRequester as sr
ON
sr.RequesterID = r.RequesterID
LEFT OUTER JOIN
SubRequesterAttribute AS sa1
ON
sa1.SubRequesterID = sr.SubRequesterID
AND
sa1.Attribute = 'Urgent'
LEFT OUTER JOIN
SubRequesterAttribute AS sa2
ON
sa2.SubRequesterID = sr.SubRequesterID
AND
sa2.Attribute = 'Closed'