It's the soul that sins
A general NT theme is of the flesh being sinful/the source of sin. E.g.:
KJV: Rom 7:18. For I know that in me (that is, in my flesh,) dwelleth no good thing: for to will is present with me; but how to perform that which is good I find not.
Rom 8:13. For if ye live after the flesh, ye shall die: but if ye through the Spirit do mortify the deeds of the body, ye shall live.
In biological terms, though, it isn't actually the hand, or foot that controls the brain, but vice-versa. It is most likely the mind/will etc. (soul) that causes us to sin, not a part of our body. Ezekiel 18 says it is "the SOUL that sins" which shall die.
There is rather a large body of Scripture (I'm still plodding thru the NT trying to understand this) seemingly about 'salvation of spirit/salvation of soul', where the first is by grace, of faith gifted by God, based on Christ's finished work on the cross. To this, we cannot add anything. The latter is arguably indicated by large parts of the NT commanding believers to stop sinning, and also I Thess 5:23 ("sanctify you wholly"? am I only partly sanctified?), and I Peter 1:9 is quite clear.
It seems that the believer's spirit probably has eternal security, since whatsoever* is born of God does not sin, and cannot sin, for 'born of God' (I John 3:9). (* STRONGS: G3956 pas includes the possible meaning "whatsoever"). Since the wages of sin is death, and the spirit born of God's Spirit (John 3:6) does/can not sin, it seems the spirit is assuredly saved. "Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it." (Eccl 12:7) This is, to my simple study, borne out by Christ Himself, Who commended His spirit into the hands of the Father (Luke 23:46) but His SOUL suffered for the world's sins that He bore:
(Now that he ascended, what is it but that he also descended first into the lower parts of the earth?(Ephesians 4:9, KJV)
KJV: Acts 2  For David speaketh concerning him [...]
 Because thou wilt not leave my soul in hell, neither wilt thou suffer thine Holy One to see corruption.
Why should the soul "die"? Because it makes the decision whether to sin or not -- in fact, can sin without the body committing a physical act. In Matt 5:28, the Lord clearly says the sin is ALREADY committed in heart/thought/soul, before the man has even had a chance to "make a pass" at the woman. In Matt 23:25-26, again: cleansing should be of/from the soul inside, and it will automatically result in clean "exterior" behavior. Merely eliminating physical acts of sin won't excuse us from judgment if we go on sinning in mind/heart without even trying to renew our minds. Judgment will include "the counsels of the hearts" (1 Cor 4:5, KJV).
Not literal amputation
Thus, it seems hardly likely that the Lord meant physical amputation of limbs, since:
a) They don't govern our actions;
b) It's still possible to sin the very worst sins in mind and heart, even if limbs were physically amputated.
So then we look for figurative or spiritual interpretation of the Lord's command. Some of my ideas follow.
Parts symbolic of specific sins?
Could it be various sins that are symbolically represented by particular body parts? Scripture mentions an "evil eye". Considering 'works of their hands', idolatry (II Kings 22:17, II Chron 34:25, Jer 1:16, Acts 7:41 etc.); violence (Isaiah 59:6, Jonah 3:8). As for feet, Proverbs 1:16, Isaiah 59:7, Jeremiah 14:10. Perhaps the Lord is parabolically telling us to seek out especially such sins and repent from them?
Church members -- parts of Christ's Body?
Rom 12:5. So we, being many, are one body in Christ, and every one members one of another. (Also I Cor 12, Eph 5:30).
We note in 1 Cor 5 the Spirit's rebuke to the Corinthians, concerning the man who "had his father's wife". Verse 2 says they should rather have mourned and prayed that this member be "taken away from among you". Verse 13 commands, "Therefore put away from among yourselves that wicked person".
Clearly, then, we have here a cutting off/throwing away of a member/part of the body of Christ, because of his flagrant sin.
a) To avoid the spread of poison to the rest of the body
(1 Cor 5:6) "a little leaven leaveneth the whole lump".
Was the Lord, in Matt 18, telling the church to cast out obvious sinners, to "cut them out of the body", lest the body be corrupted by the apparent acceptance of the sin by the elders/those who had the rule over them?
b) To avoid their own condemnation.
"For what have I to do with judging outsiders? Is it not those inside the church whom you are to judge?" (I Cor 5:12, ESV) and "[...] Happy is he that condemneth not himself in that thing which he alloweth."(Rom 14:22, KJV)
If so, then (Matt 18:9) it is better to cut out one, or three flagrant sinners, even if functioning/vital members of the assembly like an eye/hand, rather than risk corrupting the body and/or earning condemnation.