First, you can simplify your query slightly by referring to tblSerialNumber instead of web40replicensekey.
Second, LANSweeper collects what serial numbers it does separately from installed software. If you want to see the software name, version, etc. you need to link against tblSoftware and tblSoftwareUni. The serial numbers are stored separately in tblSerianNumber.
tblSoftware and tblSerialNumber can each be linked against a particular machine/asset, but there isn't anything sufficiently unique in tblSerialNumber to directly link against any particular entry in tblSoftware. The best you could do is make assumptions and fake some connections.
Quick-and-dirty:
SELECT DISTINCT TOP 1000000
tblAssets.AssetID,
tblAssets.AssetUnique,
tblAssets.Domain,
tblADusers.Firstname,
tblADusers.Lastname,
tblADusers.Department,
tblSoftwareUni.softwareName,
tblSoftware.softwareVersion,
tblSerialnumber.ProductKey
FROM
tblAssets
LEFT JOIN tblSoftware ON tblAssets.AssetID = tblSoftware.AssetID
LEFT JOIN tblSoftwareUni ON tblSoftwareUni.SoftID = tblSoftware.softID
LEFT JOIN tblSerialnumber ON tblAssets.AssetID = tblSerialnumber.AssetID AND tblSerialnumber.Product LIKE 'Adobe Acrobat%'
INNER JOIN tblAssetCustom ON tblAssets.AssetID = tblAssetCustom.AssetID
INNER JOIN tsysOS ON tblAssets.OScode = tsysOS.OScode
INNER JOIN tblADusers ON tblADusers.Username = tblAssets.Username
WHERE
tblSoftwareUni.softwareName LIKE 'Adobe Acrobat%'
With my inventory, that produces a list of Adobe Acrobats -- 8, 9 and XI -- and license keys for the 8 & 9. Presumably LANSweeper 5.1 doesn't root out keys for XI.